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Talk:UWC Red Cross Nordic

Coordinates
Hey, can someone add the coordinates for the college? I've done it for Mahindra College already, would nice to have them for all colleges (so people like me can finally find out where RCN actually is :-)) How to go about it: 1) Find the college in something like Google Earth and note down the coordinate 2) Put a tag of the type 18.52472°N, 73.58528°W in the Wiki text, it will result in something like this: 18.52472°N, 73.58528°W - which leads users to a page where they can select a map and look at it. Cheers, Tobi Tobi Kellner -- The co-ordinates are 61.334241,5.334654, I don't know how to insert them. —Preceding unsigned comment added by 158.143.181.110 (talk) 13:14, 9 February 2010 (UTC)

Number of Students
Not all colleges have 200 students, notable exceptions being the College of the Atlantic and the College of South East Asia. Rune Welsh 21:46, Apr 19, 2005 (UTC)

Image copyright problem with Image:Official logo uwc.png
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title: Talk:UWC Red Cross Nordic


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User:Vince Vatter

I am a professor of mathematics at the University of Florida.

Some links

 * Cite your sources for templates and things
 * User:MathMartin has other good links
 * Mathematics Collaboration of the Week
 * WikiProject Mathematics
 * User:Charles Matthews/notation
 * Pages needing attention/Mathematics

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Talk:Pölsa

Create a page for Pølse?
Pölsa and Pølse, while related words are two different languages and two different things. If this ;

> The word pølse can also describe the appearance of an object, like "it looks like a pølse", and "it's pølse shaped" (i.e. sausage shaped).[7]

is important to have in an encyclopedia shouldn't pølse have it's own page? 31.211.192.162 (talk) 05:52, 7 May 2016 (UTC)

..
Just if you want to know, this stuff tastes like crap, don't try it. (unsigned)

 * Rubbish! It's just as god as haggis or any other kind of wholesome, traditionally rural food made from delicious low grade slaughter by-products. Peter Isotalo 15:37, Apr 26, 2005 (UTC)

 * I’m usually quite a picky eater, but i can eat pölsa. I can’t say I like it, but to me it’s like any other kind of normal type of food.  I suppose it’s possible you have to be raised on the stuff to stomach it.
 * --194.47.143.2 21:51, 19 April 2006 (UTC)

 * I love pölsa, though I agree it's probably an acquired taste.

 * If I ever come to Sverige I'll probably try it, especially if I can get it without liver. How does it go with grits? (By the way: fixed the image link so it displays)--Orange Mike 18:34, 6 November 2006 (UTC)

WikiProject Food and drink Tagging
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External links modified
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Matrix calculus

In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics.

Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors). A single convention can be somewhat standard throughout a single field that commonly uses matrix calculus (e.g. econometrics, statistics, estimation theory and machine learning). However, even within a given field different authors can be found using competing conventions. Authors of both groups often write as though their specific conventions were standard. Serious mistakes can result when combining results from different authors without carefully verifying that compatible notations have been used. Definitions of these two conventions and comparisons between them are collected in the layout conventions section.

Scope
Matrix calculus refers to a number of different notations that use matrices and vectors to collect the derivative of each component of the dependent variable with respect to each component of the independent variable. In general, the independent variable can be a scalar, a vector, or a matrix while the dependent variable can be any of these as well. Each different situation will lead to a different set of rules, or a separate calculus, using the broader sense of the term. Matrix notation serves as a convenient way to collect the many derivatives in an organized way.

As a first example, consider the gradient from vector calculus. For a scalar function of three independent variables, $$f(x_1, x_2, x_3)$$, the gradient is given by the vector equation
 * $$\nabla f = \frac{\partial f}{\partial x_1} \hat{x}_1 + \frac{\partial f}{\partial x_2} \hat{x}_2 + \frac{\partial f}{\partial x_3} \hat{x}_3 ,$$

where $$\hat{x}_i$$ represents a unit vector in the $$x_i$$ direction for $$1\le i \le 3$$. This type of generalized derivative can be seen as the derivative of a scalar, f, with respect to a vector, $$\mathbf{x}$$, and its result can be easily collected in vector form.
 * $$\nabla f = \left( \frac{\partial f}{\partial \mathbf{x}} \right)^{\mathsf{T}} =

\begin{bmatrix} \dfrac{\partial f}{\partial x_1} & \dfrac{\partial f}{\partial x_2} & \dfrac{\partial f}{\partial x_3} \\ \end{bmatrix}^\textsf{T}. $$

More complicated examples include the derivative of a scalar function with respect to a matrix, known as the gradient matrix, which collects the derivative with respect to each matrix element in the corresponding position in the resulting matrix. In that case the scalar must be a function of each of the independent variables in the matrix. As another example, if we have an $n$-vector of dependent variables, or functions, of $m$ independent variables we might consider the derivative of the dependent vector with respect to the independent vector. The result could be collected in an $m×n$ matrix consisting of all of the possible derivative combinations.

There are a total of nine possibilities using scalars, vectors, and matrices. Notice that as we consider higher numbers of components in each of the independent and dependent variables we can be left with a very large number of possibilities. The six kinds of derivatives that can be most neatly organized in matrix form are collected in the following table.

Here, we have used the term "matrix" in its most general sense, recognizing that vectors are simply matrices with one column (and scalars are simply vectors with one row). Moreover, we have used bold letters to indicate vectors and bold capital letters for matrices. This notation is used throughout.

Notice that we could also talk about the derivative of a vector with respect to a matrix, or any of the other unfilled cells in our table. However, these derivatives are most naturally organized in a tensor of rank higher than 2, so that they do not fit neatly into a matrix. In the following three sections we will define each one of these derivatives and relate them to other branches of mathematics. See the layout conventions section for a more detailed table.

Relation to other derivatives
The matrix derivative is a convenient notation for keeping track of partial derivatives for doing calculations. The Fréchet derivative is the standard way in the setting of functional analysis to take derivatives with respect to vectors. In the case that a matrix function of a matrix is Fréchet differentiable, the two derivatives will agree up to translation of notations. As is the case in general for partial derivatives, some formulae may extend under weaker analytic conditions than the existence of the derivative as approximating linear mapping.

Usages
Matrix calculus is used for deriving optimal stochastic estimators, often involving the use of Lagrange multipliers. This includes the derivation of:
 * Kalman filter
 * Wiener filter
 * Expectation-maximization algorithm for Gaussian mixture
 * Gradient descent

Notation
The vector and matrix derivatives presented in the sections to follow take full advantage of matrix notation, using a single variable to represent a large number of variables. In what follows we will distinguish scalars, vectors and matrices by their typeface. We will let $M(n,m)$ denote the space of real $n×m$ matrices with $n$ rows and $m$ columns. Such matrices will be denoted using bold capital letters: $A$, $X$, $Y$, etc. An element of $M(n,1)$, that is, a column vector, is denoted with a boldface lowercase letter: $a$, $x$, $y$, etc. An element of $M(1,1)$ is a scalar, denoted with lowercase italic typeface: $a$, $t$, $x$, etc. $X^{T}$ denotes matrix transpose, $tr(X)$ is the trace, and $det(X)$ or $|X|$ is the determinant. All functions are assumed to be of differentiability class $C^{1}$ unless otherwise noted. Generally letters from the first half of the alphabet (a, b, c, ...) will be used to denote constants, and from the second half (t, x, y, ...) to denote variables.

NOTE: As mentioned above, there are competing notations for laying out systems of partial derivatives in vectors and matrices, and no standard appears to be emerging yet. The next two introductory sections use the numerator layout convention simply for the purposes of convenience, to avoid overly complicating the discussion. The section after them discusses layout conventions in more detail. It is important to realize the following:
 * 1) Despite the use of the terms "numerator layout" and "denominator layout", there are actually more than two possible notational choices involved. The reason is that the choice of numerator vs. denominator (or in some situations, numerator vs. mixed) can be made independently for scalar-by-vector, vector-by-scalar, vector-by-vector, and scalar-by-matrix derivatives, and a number of authors mix and match their layout choices in various ways.
 * 2) The choice of numerator layout in the introductory sections below does not imply that this is the "correct" or "superior" choice. There are advantages and disadvantages to the various layout types.  Serious mistakes can result from carelessly combining formulas written in different layouts, and converting from one layout to another requires care to avoid errors.  As a result, when working with existing formulas the best policy is probably to identify whichever layout is used and maintain consistency with it, rather than attempting to use the same layout in all situations.

Alternatives
The tensor index notation with its Einstein summation convention is very similar to the matrix calculus, except one writes only a single component at a time. It has the advantage that one can easily manipulate arbitrarily high rank tensors, whereas tensors of rank higher than two are quite unwieldy with matrix notation. All of the work here can be done in this notation without use of the single-variable matrix notation. However, many problems in estimation theory and other areas of applied mathematics would result in too many indices to properly keep track of, pointing in favor of matrix calculus in those areas. Also, Einstein notation can be very useful in proving the identities presented here (see section on differentiation) as an alternative to typical element notation, which can become cumbersome when the explicit sums are carried around. Note that a matrix can be considered a tensor of rank two.

Derivatives with vectors
Because vectors are matrices with only one column, the simplest matrix derivatives are vector derivatives.

The notations developed here can accommodate the usual operations of vector calculus by identifying the space $M(n,1)$ of $n$-vectors with the Euclidean space $R^{n}$, and the scalar $M(1,1)$ is identified with $R$. The corresponding concept from vector calculus is indicated at the end of each subsection.

NOTE: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.

Vector-by-scalar
The derivative of a vector $$ \mathbf{y} = \begin{bmatrix} y_1 & y_2 & \cdots & y_m \end{bmatrix}^\mathsf{T} $$, by a scalar $x$ is written (in numerator layout notation) as

 * $$\frac{d \mathbf{y}}{d x} =

\begin{bmatrix} \frac{d y_1}{d x}\\ \frac{d y_2}{d x}\\ \vdots\\ \frac{d y_m}{d x}\\ \end{bmatrix}. $$

In vector calculus the derivative of a vector $y$ with respect to a scalar $x$ is known as the tangent vector of the vector $y$, $$\frac{\partial \mathbf{y}}{\partial x}$$. Notice here that $y: R^{1} → R^{m}$.

Example Simple examples of this include the velocity vector in Euclidean space, which is the tangent vector of the position vector (considered as a function of time). Also, the acceleration is the tangent vector of the velocity.

Scalar-by-vector
The derivative of a scalar $y$ by a vector $$ \mathbf{x} = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^\mathsf{T} $$, is written (in numerator layout notation) as

 * $$\frac{\partial y}{\partial \mathbf{x}} =

\begin{bmatrix} \dfrac{\partial y}{\partial x_1} & \dfrac{\partial y}{\partial x_2} & \cdots & \dfrac{\partial y}{\partial x_n} \end{bmatrix}. $$

In vector calculus, the gradient of a scalar field $f : R^{n} → R$ (whose independent coordinates are the components of $x$) is the transpose of the derivative of a scalar by a vector.

 * $$\nabla f = \begin{bmatrix}\frac{\partial f}{\partial x_1} \\ \vdots \\ \frac{\partial f}{\partial x_n} \end{bmatrix} = \left( \frac{\partial f}{\partial \mathbf{x}} \right)^{\mathsf{T}}$$

By example, in physics, the electric field is the negative vector gradient of the electric potential.

The directional derivative of a scalar function $f(x)$ of the space vector $x$ in the direction of the unit vector $u$ (represented in this case as a column vector) is defined using the gradient as follows.
 * $$\nabla_{\mathbf{u}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot \mathbf{u}$$

Using the notation just defined for the derivative of a scalar with respect to a vector we can re-write the directional derivative as $$\nabla_\mathbf{u} f = \frac{\partial f}{\partial \mathbf{x}} \mathbf{u}.$$ This type of notation will be nice when proving product rules and chain rules that come out looking similar to what we are familiar with for the scalar derivative.

Vector-by-vector
Each of the previous two cases can be considered as an application of the derivative of a vector with respect to a vector, using a vector of size one appropriately. Similarly we will find that the derivatives involving matrices will reduce to derivatives involving vectors in a corresponding way.

The derivative of a vector function (a vector whose components are functions) $$ \mathbf{y} = \begin{bmatrix} y_1 & y_2 & \cdots & y_m \end{bmatrix}^\mathsf{T} $$, with respect to an input vector, $$ \mathbf{x} = \begin{bmatrix} x_1 & x_2 & \cdots & x_n \end{bmatrix}^\mathsf{T} $$, is written (in numerator layout notation) as

 * $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}} =

\begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n}\\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n}\\ \end{bmatrix}. $$

In vector calculus, the derivative of a vector function $y$ with respect to a vector $x$ whose components represent a space is known as the pushforward (or differential), or the Jacobian matrix.

The pushforward along a vector function $f$ with respect to vector $v$ in $R^{n}$ is given by $$ d\mathbf{f}(\mathbf{v}) = \frac{\partial \mathbf{f}}{\partial \mathbf{v}} d\mathbf{v}. $$

Derivatives with matrices
There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix. These can be useful in minimization problems found in many areas of applied mathematics and have adopted the names tangent matrix and gradient matrix respectively after their analogs for vectors.

Note: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.

Matrix-by-scalar
The derivative of a matrix function $Y$ by a scalar $x$ is known as the tangent matrix and is given (in numerator layout notation) by

\frac{\partial \mathbf{Y}}{\partial x} = \begin{bmatrix} \frac{\partial y_{11}}{\partial x} & \frac{\partial y_{12}}{\partial x} & \cdots & \frac{\partial y_{1n}}{\partial x}\\ \frac{\partial y_{21}}{\partial x} & \frac{\partial y_{22}}{\partial x} & \cdots & \frac{\partial y_{2n}}{\partial x}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y_{m1}}{\partial x} & \frac{\partial y_{m2}}{\partial x} & \cdots & \frac{\partial y_{mn}}{\partial x}\\ \end{bmatrix}. $$

Scalar-by-matrix
The derivative of a scalar function $y$, with respect to a $p×q$ matrix $X$ of independent variables, is given (in numerator layout notation) by

\frac{\partial y}{\partial \mathbf{X}} = \begin{bmatrix} \frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{21}} & \cdots & \frac{\partial y}{\partial x_{p1}}\\ \frac{\partial y}{\partial x_{12}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{p2}}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y}{\partial x_{1q}} & \frac{\partial y}{\partial x_{2q}} & \cdots & \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}. $$

Important examples of scalar functions of matrices include the trace of a matrix and the determinant.

In analog with vector calculus this derivative is often written as the following.

\nabla_\mathbf{X} y(\mathbf{X}) = \frac{\partial y(\mathbf{X})}{\partial \mathbf{X}} $$ Also in analog with vector calculus, the directional derivative of a scalar $f(X)$ of a matrix $X$ in the direction of matrix $Y$ is given by
 * $$\nabla_\mathbf{Y} f = \operatorname{tr} \left(\frac{\partial f}{\partial \mathbf{X}} \mathbf{Y}\right).$$

It is the gradient matrix, in particular, that finds many uses in minimization problems in estimation theory, particularly in the derivation of the Kalman filter algorithm, which is of great importance in the field.

Other matrix derivatives
The three types of derivatives that have not been considered are those involving vectors-by-matrices, matrices-by-vectors, and matrices-by-matrices. These are not as widely considered and a notation is not widely agreed upon.

Layout conventions
This section discusses the similarities and differences between notational conventions that are used in the various fields that take advantage of matrix calculus. Although there are largely two consistent conventions, some authors find it convenient to mix the two conventions in forms that are discussed below. After this section, equations will be listed in both competing forms separately.

The fundamental issue is that the derivative of a vector with respect to a vector, i.e. $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$$, is often written in two competing ways. If the numerator $y$ is of size $m$ and the denominator $x$ of size n, then the result can be laid out as either an $m×n$ matrix or $n×m$ matrix, i.e. the $m$ elements of $y$ laid out in rows and the $n$ elements of $x$ laid out in columns, or vice versa. This leads to the following possibilities:
 * 1) Numerator layout, i.e. lay out according to $y$ and $x^{T}$ (i.e. contrarily to $x$).  This is sometimes known as the Jacobian formulation. This corresponds to the $m×n$ layout in the previous example, which means that the row number of $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$$ equals to the size of the numerator $$\mathbf{y}$$ and the column number of $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$$ equals to the size of $x^{T}$.
 * 2) Denominator layout, i.e. lay out according to $y^{T}$ and $x$ (i.e. contrarily to y). This is sometimes known as the Hessian formulation.  Some authors term this layout the gradient, in distinction to the Jacobian (numerator layout), which is its transpose. (However, gradient more commonly means the derivative $$\frac{\partial y}{\partial \mathbf{x}},$$ regardless of layout.). This corresponds to the n×m layout in the previous example, which means that the row number of $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$$ equals to the size of $x$ (the denominator).
 * 3) A third possibility sometimes seen is to insist on writing the derivative as $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}'},$$ (i.e. the derivative is taken with respect to the transpose of $x$) and follow the numerator layout. This makes it possible to claim that the matrix is laid out according to both numerator and denominator.  In practice this produces results the same as the numerator layout.

When handling the gradient $$\frac{\partial y}{\partial \mathbf{x}}$$ and the opposite case $$\frac{\partial \mathbf{y}}{\partial x},$$ we have the same issues. To be consistent, we should do one of the following:
 * 1) If we choose numerator layout for $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}},$$ we should lay out the gradient $$\frac{\partial y}{\partial \mathbf{x}}$$ as a row vector, and $$\frac{\partial \mathbf{y}}{\partial x}$$ as a column vector.
 * 2) If we choose denominator layout for $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}},$$ we should lay out the gradient $$\frac{\partial y}{\partial \mathbf{x}}$$ as a column vector, and $$\frac{\partial \mathbf{y}}{\partial x}$$ as a row vector.
 * 3) In the third possibility above, we write $$\frac{\partial y}{\partial \mathbf{x}'}$$ and $$\frac{\partial \mathbf{y}}{\partial x},$$ and use numerator layout.

Not all math textbooks and papers are consistent in this respect throughout. That is, sometimes different conventions are used in different contexts within the same book or paper. For example, some choose denominator layout for gradients (laying them out as column vectors), but numerator layout for the vector-by-vector derivative $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}.$$

Similarly, when it comes to scalar-by-matrix derivatives $$\frac{\partial y}{\partial \mathbf{X}}$$ and matrix-by-scalar derivatives $$\frac{\partial \mathbf{Y}}{\partial x},$$ then consistent numerator layout lays out according to $Y$ and $X^{T}$, while consistent denominator layout lays out according to $Y^{T}$ and $X$. In practice, however, following a denominator layout for $$\frac{\partial \mathbf{Y}}{\partial x},$$ and laying the result out according to $Y^{T}$, is rarely seen because it makes for ugly formulas that do not correspond to the scalar formulas. As a result, the following layouts can often be found:
 * 1) Consistent numerator layout, which lays out $$\frac{\partial \mathbf{Y}}{\partial x}$$ according to $Y$ and $$\frac{\partial y}{\partial \mathbf{X}}$$ according to $X^{T}$.
 * 2) Mixed layout, which lays out $$\frac{\partial \mathbf{Y}}{\partial x}$$ according to $Y$ and $$\frac{\partial y}{\partial \mathbf{X}}$$ according to $X$.
 * 3) Use the notation $$\frac{\partial y}{\partial \mathbf{X}'},$$ with results the same as consistent numerator layout.

In the following formulas, we handle the five possible combinations $$\frac{\partial y}{\partial \mathbf{x}}, \frac{\partial \mathbf{y}}{\partial x}, \frac{\partial \mathbf{y}}{\partial \mathbf{x}}, \frac{\partial y}{\partial \mathbf{X}}$$ and $$\frac{\partial \mathbf{Y}}{\partial x}$$ separately. We also handle cases of scalar-by-scalar derivatives that involve an intermediate vector or matrix. (This can arise, for example, if a multi-dimensional parametric curve is defined in terms of a scalar variable, and then a derivative of a scalar function of the curve is taken with respect to the scalar that parameterizes the curve.) For each of the various combinations, we give numerator-layout and denominator-layout results, except in the cases above where denominator layout rarely occurs. In cases involving matrices where it makes sense, we give numerator-layout and mixed-layout results. As noted above, cases where vector and matrix denominators are written in transpose notation are equivalent to numerator layout with the denominators written without the transpose.

Keep in mind that various authors use different combinations of numerator and denominator layouts for different types of derivatives, and there is no guarantee that an author will consistently use either numerator or denominator layout for all types. Match up the formulas below with those quoted in the source to determine the layout used for that particular type of derivative, but be careful not to assume that derivatives of other types necessarily follow the same kind of layout.

When taking derivatives with an aggregate (vector or matrix) denominator in order to find a maximum or minimum of the aggregate, it should be kept in mind that using numerator layout will produce results that are transposed with respect to the aggregate. For example, in attempting to find the maximum likelihood estimate of a multivariate normal distribution using matrix calculus, if the domain is a k×1 column vector, then the result using the numerator layout will be in the form of a 1×k row vector. Thus, either the results should be transposed at the end or the denominator layout (or mixed layout) should be used.

 * {|class="wikitable"

! colspan=2 rowspan=2 | ! colspan=2 | Scalar $y$ ! colspan=2 | Column vector $y$ (size $m×1$) ! colspan=2 | Matrix $Y$ (size $m×n$) ! Notation !! Type ! Notation !! Type ! Notation !! Type ! rowspan=2 | Scalar $x$ ! Numerator ! Denominator ! rowspan=2 | Column vector $m×n$ (size $x$) ! Numerator ! Denominator ! rowspan=2 | Matrix $n×1$ (size $m×n$) ! Numerator ! Denominator
 * + Result of differentiating various kinds of aggregates with other kinds of aggregates
 * rowspan=2 style="text-align:center;" | $$\frac{\partial y}{\partial x}$$
 * rowspan=2 | Scalar
 * rowspan=2 style="text-align:center;" | $$\frac{\partial \mathbf{y}}{\partial x}$$
 * Size-$m$ column vector
 * rowspan=2 style="text-align:center;" | $$\frac{\partial \mathbf{Y}}{\partial x}$$
 * $n×m$ matrix
 * Size-m row vector
 * rowspan=2 style="text-align:center;" | $$\frac{\partial y}{\partial \mathbf{x}}$$
 * Size-$n$ row vector
 * rowspan=2 style="text-align:center;" | $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$$
 * $X$ matrix
 * rowspan=2 style="text-align:center;" | $$\frac{\partial \mathbf{Y}}{\partial \mathbf{x}}$$
 * rowspan=2 |
 * Size-$n$ column vector
 * $p×q$ matrix
 * rowspan=2 style="text-align:center;" | $$\frac{\partial y}{\partial \mathbf{X}}$$
 * $q×p$ matrix
 * rowspan=2 style="text-align:center;" | $$\frac{\partial \mathbf{y}}{\partial \mathbf{X}}$$
 * rowspan=2 |
 * rowspan=2 style="text-align:center;" | $$\frac{\partial \mathbf{Y}}{\partial \mathbf{X}}$$
 * rowspan=2 |
 * $p×q$ matrix
 * }

The results of operations will be transposed when switching between numerator-layout and denominator-layout notation.

Numerator-layout notation
Using numerator-layout notation, we have:

 * $$\begin{align}

\frac{\partial y}{\partial \mathbf{x}} &= \begin{bmatrix} \frac{\partial y}{\partial x_1} & \frac{\partial y}{\partial x_2} & \cdots                         & \frac{\partial y}{\partial x_n} \end{bmatrix}. \\ \frac{\partial \mathbf{y}}{\partial x} &= \begin{bmatrix} \frac{\partial y_1}{\partial x} \\ \frac{\partial y_2}{\partial x} \\ \vdots                         \\ \frac{\partial y_m}{\partial x} \\ \end{bmatrix}. \\ \frac{\partial \mathbf{y}}{\partial \mathbf{x}} &= \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n} \\ \vdots                           & \vdots                            & \ddots & \vdots \\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n} \\ \end{bmatrix}. \\ \frac{\partial y}{\partial \mathbf{X}} &= \begin{bmatrix} \frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{21}} & \cdots & \frac{\partial y}{\partial x_{p1}} \\ \frac{\partial y}{\partial x_{12}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{p2}} \\ \vdots                            & \vdots                             & \ddots & \vdots \\ \frac{\partial y}{\partial x_{1q}} & \frac{\partial y}{\partial x_{2q}} & \cdots & \frac{\partial y}{\partial x_{pq}} \\ \end{bmatrix}. \end{align}$$

The following definitions are only provided in numerator-layout notation:

 * $$\begin{align}

\frac{\partial \mathbf{Y}}{\partial x} &= \begin{bmatrix} \frac{\partial y_{11}}{\partial x} & \frac{\partial y_{12}}{\partial x} & \cdots & \frac{\partial y_{1n}}{\partial x} \\ \frac{\partial y_{21}}{\partial x} & \frac{\partial y_{22}}{\partial x} & \cdots & \frac{\partial y_{2n}}{\partial x} \\ \vdots                            & \vdots                             & \ddots & \vdots \\ \frac{\partial y_{m1}}{\partial x} & \frac{\partial y_{m2}}{\partial x} & \cdots & \frac{\partial y_{mn}}{\partial x} \\ \end{bmatrix}. \\ d\mathbf{X} &= \begin{bmatrix} dx_{11} & dx_{12} & \cdots & dx_{1n} \\ dx_{21} & dx_{22} & \cdots & dx_{2n} \\ \vdots & \vdots  & \ddots & \vdots \\ dx_{m1} & dx_{m2} & \cdots & dx_{mn} \\ \end{bmatrix}. \end{align}$$

Denominator-layout notation
Using denominator-layout notation, we have:
 * $$\begin{align}

\frac{\partial y}{\partial \mathbf{x}} &= \begin{bmatrix} \frac{\partial y}{\partial x_1}\\ \frac{\partial y}{\partial x_2}\\ \vdots\\ \frac{\partial y}{\partial x_n}\\ \end{bmatrix}. \\ \frac{\partial \mathbf{y}}{\partial x} &= \begin{bmatrix} \frac{\partial y_1}{\partial x} & \frac{\partial y_2}{\partial x} & \cdots & \frac{\partial y_m}{\partial x} \end{bmatrix}. \\ \frac{\partial \mathbf{y}}{\partial \mathbf{x}} &= \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_2}{\partial x_1} & \cdots & \frac{\partial y_m}{\partial x_1} \\ \frac{\partial y_1}{\partial x_2} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_2} \\ \vdots                           & \vdots                            & \ddots & \vdots \\ \frac{\partial y_1}{\partial x_n} & \frac{\partial y_2}{\partial x_n} & \cdots & \frac{\partial y_m}{\partial x_n}\\ \end{bmatrix}. \\ \frac{\partial y}{\partial \mathbf{X}} &= \begin{bmatrix} \frac{\partial y}{\partial x_{11}} & \frac{\partial y}{\partial x_{12}} & \cdots & \frac{\partial y}{\partial x_{1q}}\\ \frac{\partial y}{\partial x_{21}} & \frac{\partial y}{\partial x_{22}} & \cdots & \frac{\partial y}{\partial x_{2q}}\\ \vdots                            & \vdots                             & \ddots & \vdots\\ \frac{\partial y}{\partial x_{p1}} & \frac{\partial y}{\partial x_{p2}} & \cdots & \frac{\partial y}{\partial x_{pq}}\\ \end{bmatrix}. \end{align}$$

Identities
As noted above, in general, the results of operations will be transposed when switching between numerator-layout and denominator-layout notation.

To help make sense of all the identities below, keep in mind the most important rules: the chain rule, product rule and sum rule. The sum rule applies universally, and the product rule applies in most of the cases below, provided that the order of matrix products is maintained, since matrix products are not commutative. The chain rule applies in some of the cases, but unfortunately does not apply in matrix-by-scalar derivatives or scalar-by-matrix derivatives (in the latter case, mostly involving the trace operator applied to matrices). In the latter case, the product rule can't quite be applied directly, either, but the equivalent can be done with a bit more work using the differential identities.

The following identities adopt the following conventions:
 * the scalars, $a$, $b$, $c$, $d$, and $e$ are constant in respect of, and the scalars, $u$, and $v$ are functions of one of $x$, $x$, or $X$;
 * the vectors, $a$, $b$, $c$, $d$, and $e$ are constant in respect of, and the vectors, $u$, and $v$ are functions of one of $x$, $x$, or $X$;
 * the matrices, $A$, $B$, $C$, $D$, and $E$ are constant in respect of, and the matrices, $U$ and $V$ are functions of one of $x$, $x$, or $X$.

Vector-by-vector identities
This is presented first because all of the operations that apply to vector-by-vector differentiation apply directly to vector-by-scalar or scalar-by-vector differentiation simply by reducing the appropriate vector in the numerator or denominator to a scalar.

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="150" | Condition ! scope="col" width="10" | Expression ! scope="col" width="100" | Numerator layout, i.e. by $y$ and $x^{T}$ ! scope="col" width="100" | Denominator layout, i.e. by $y^{T}$ and $x$ $a$ || $$\frac{\partial a\mathbf{u}}{\partial\, \mathbf{x}} =$$ $x$ is not a function of $A$ || $$\frac{\partial v\mathbf{a}}{\partial \mathbf{x}} =$$ || $$\mathbf{a}\frac{\partial v}{\partial \mathbf{x}} $$ || $$\frac{\partial v}{\partial \mathbf{x}} \mathbf{a}^\top$$ $x$ || $$\frac{\partial \mathbf{A}\mathbf{u}}{\partial \mathbf{x}} =$$ || $$\mathbf{A}\frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$ || $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}\mathbf{A}^\top$$
 * + Identities: vector-by-vector $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}}$$
 * $A$ is not a function of $x$ || $$\frac{\partial \mathbf{a}}{\partial \mathbf{x}} =$$ ||colspan=2| $$\mathbf{0}$$
 * || $$\frac{\partial \mathbf{x}}{\partial \mathbf{x}} =$$ || colspan=2|$$\mathbf{I}$$
 * $a$ is not a function of $x$ || $$\frac{\partial \mathbf{A}\mathbf{x}}{\partial \mathbf{x}} =$$ || $$\mathbf{A}$$ || $$\mathbf{A}^\top$$
 * $u = u(x)$ is not a function of $v = v(x)$ || $$\frac{\partial \mathbf{x}^\top \mathbf{A}}{\partial \mathbf{x}} =$$ || $$\mathbf{A}^\top$$ || $$\mathbf{A}$$
 * $a$ is not a function of $x$,
 * $v = v(x)$ is not a function of $u = u(x)$ || $$\frac{\partial \mathbf{A}\mathbf{x}}{\partial \mathbf{x}} =$$ || $$\mathbf{A}$$ || $$\mathbf{A}^\top$$
 * $A$ is not a function of $x$ || $$\frac{\partial \mathbf{x}^\top \mathbf{A}}{\partial \mathbf{x}} =$$ || $$\mathbf{A}^\top$$ || $$\mathbf{A}$$
 * $u = u(x)$ is not a function of $u = u(x)$,
 * $v = v(x)$ is not a function of $u = u(x)$,
 * $u = u(x)$ is not a function of $x^{T}$,
 * colspan=2|$$a\frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$
 * $x$, $a$ || $$\frac{\partial v\mathbf{u}}{\partial \mathbf{x}} =$$ || $$v \frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \mathbf{u}\frac{\partial v}{\partial \mathbf{x}} $$ || $$v\frac{\partial \mathbf{u}}{\partial \mathbf{x}}  +  \frac{\partial v}{\partial \mathbf{x}} \mathbf{u}^\top$$
 * $x$ is not a function of $x$,
 * $a$ is not a function of $x$,
 * $u = u(x)$ is not a function of $u = u(x)$,
 * $v = v(x)$, $u = u(x)$ || $$\frac{\partial (\mathbf{u} + \mathbf{v})}{\partial \mathbf{x}} =$$
 * colspan=2|$$\frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$
 * $v = v(x)$ || $$\frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{x}} =$$|| $$\frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$|| $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}}$$
 * $u = u(x)$ || $$\frac{\partial \mathbf{f}(\mathbf{g}(\mathbf{u}))}{\partial \mathbf{x}} =$$|| $$\frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$|| $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}}$$
 * }
 * $u = u(x)$ || $$\frac{\partial \mathbf{f}(\mathbf{g}(\mathbf{u}))}{\partial \mathbf{x}} =$$|| $$\frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$|| $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}}$$
 * }
 * }

Scalar-by-vector identities
The fundamental identities are placed above the thick black line.

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="150" | Condition ! scope="col" width="200" | Expression ! scope="col" width="200" | Numerator layout, i.e. by $u = u(x)$; result is row vector ! scope="col" width="200" | Denominator layout, i.e. by $v = v(x)$; result is column vector $u = u(x)$ || $$\frac{\partial au}{\partial \mathbf{x}} =$$ $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}, \frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$ in numerator layout $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}, \frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$ in denominator layout $v = v(x)$ is not a function of $A$ $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}, \frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$ in numerator layout $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}, \frac{\partial \mathbf{v}}{\partial \mathbf{x}}$$ in denominator layout
 * + Identities: scalar-by-vector $$\frac{\partial y}{\partial \mathbf{x}} = \nabla_\mathbf{x} y$$
 * $x$ is not a function of $a$ || $$\frac{\partial a}{\partial \mathbf{x}} =$$
 * $$\mathbf{0}^\top$$ ||$$\mathbf{0}$$
 * $x$ is not a function of $A$,
 * $x$ is not a function of $b$,
 * $x$ is not a function of $A$,
 * colspan=2|$$a\frac{\partial u}{\partial \mathbf{x}}$$
 * $x$, $A$ || $$\frac{\partial (u+v)}{\partial \mathbf{x}} =$$
 * colspan=2|$$\frac{\partial u}{\partial \mathbf{x}} + \frac{\partial v}{\partial \mathbf{x}} $$
 * $x$, $A$ || $$\frac{\partial uv}{\partial \mathbf{x}} =$$
 * colspan=2|$$u\frac{\partial v}{\partial \mathbf{x}} + v\frac{\partial u}{\partial \mathbf{x}} $$
 * $A$ || $$\frac{\partial g(u)}{\partial \mathbf{x}} =$$
 * colspan=2|$$\frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf{x}} $$
 * $x$ || $$\frac{\partial f(g(u))}{\partial \mathbf{x}} =$$
 * colspan=2|$$\frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf{x}} $$
 * $$\frac{\partial (\mathbf{u} \cdot \mathbf{v})}{\partial \mathbf{x}} = \frac{\partial \mathbf{u}^\top \mathbf{v}}{\partial \mathbf{x}} =$$
 * $$\mathbf{u}^\top\frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \mathbf{v}^\top\frac{\partial \mathbf{u}}{\partial \mathbf{x}} $$
 * $A$ || $$\frac{\partial f(g(u))}{\partial \mathbf{x}} =$$
 * colspan=2|$$\frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf{x}} $$
 * $$\frac{\partial (\mathbf{u} \cdot \mathbf{v})}{\partial \mathbf{x}} = \frac{\partial \mathbf{u}^\top \mathbf{v}}{\partial \mathbf{x}} =$$
 * $$\mathbf{u}^\top\frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \mathbf{v}^\top\frac{\partial \mathbf{u}}{\partial \mathbf{x}} $$
 * $$\frac{\partial (\mathbf{u} \cdot \mathbf{v})}{\partial \mathbf{x}} = \frac{\partial \mathbf{u}^\top \mathbf{v}}{\partial \mathbf{x}} =$$
 * $$\mathbf{u}^\top\frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \mathbf{v}^\top\frac{\partial \mathbf{u}}{\partial \mathbf{x}} $$
 * $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}\mathbf{v} + \frac{\partial \mathbf{v}}{\partial \mathbf{x}}\mathbf{u}$$
 * $$\frac{\partial (\mathbf{u} \cdot \mathbf{A}\mathbf{v})}{\partial \mathbf{x}} = \frac{\partial \mathbf{u}^\top\mathbf{A}\mathbf{v}}{\partial \mathbf{x}} =$$
 * $$\mathbf{u}^\top\mathbf{A}\frac{\partial \mathbf{v}}{\partial \mathbf{x}} + \mathbf{v}^\top \mathbf{A}^\top\frac{\partial \mathbf{u}}{\partial \mathbf{x}} $$
 * $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}\mathbf{A}\mathbf{v} + \frac{\partial \mathbf{v}}{\partial \mathbf{x}}\mathbf{A}^\top\mathbf{u}$$
 * $$\frac{\partial^2 f}{\partial\mathbf{x} \partial\mathbf{x}^\top} =$$
 * $$\mathbf{H}^\top$$
 * $$\mathbf{H}$$, the Hessian matrix
 * - style="border-top: 3px solid;"
 * $x$ is not a function of $A$ || $$\frac{\partial (\mathbf{a}\cdot\mathbf{x})}{\partial \mathbf{x}} = \frac{\partial (\mathbf{x}\cdot\mathbf{a})}{\partial \mathbf{x}} =$$
 * - style="border-top: 3px solid;"
 * $a$ is not a function of $x$ || $$\frac{\partial (\mathbf{a}\cdot\mathbf{x})}{\partial \mathbf{x}} = \frac{\partial (\mathbf{x}\cdot\mathbf{a})}{\partial \mathbf{x}} =$$

$$\frac{\partial \mathbf{a}^\top\mathbf{x}}{\partial \mathbf{x}} = \frac{\partial \mathbf{x}^\top\mathbf{a}}{\partial \mathbf{x}} =$$ || $$\mathbf{a}^\top$$ || $$\mathbf{a}$$ $u = u(x)$ is not a function of $a$ || $$\frac{\partial \mathbf{b}^\top\mathbf{A}\mathbf{x}}{\partial \mathbf{x}} =$$ || $$\mathbf{b}^\top\mathbf{A}$$ || $$\mathbf{A}^\top\mathbf{b}$$ $b$ is symmetric || $$\frac{\partial \mathbf{x}^\top\mathbf{A}\mathbf{x}}{\partial \mathbf{x}} =$$ || $$2\mathbf{x}^\top\mathbf{A}$$ || $$2\mathbf{A}\mathbf{x}$$ $x$ is symmetric || $$\frac{\partial^2 \mathbf{x}^\top\mathbf{A}\mathbf{x}}{\partial\mathbf{x} \partial\mathbf{x}^\top} =$$ || colspan=2|$$2\mathbf{A}$$ $A$ $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$ in numerator layout $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$ in denominator layout
 * $b$ is not a function of $C$
 * $D$ is not a function of $e$
 * $x$ is not a function of $a$ || $$\frac{\partial \mathbf{x}^\top\mathbf{A}\mathbf{x}}{\partial \mathbf{x}} =$$ || $$\mathbf{x}^\top\left(\mathbf{A} + \mathbf{A}^\top\right)$$ || $$\left(\mathbf{A} + \mathbf{A}^\top\right)\mathbf{x}$$
 * $x$ is not a function of $y$
 * $y^{T}$ is not a function of $a$
 * $x$ is not a function of $a$
 * $x$ is not a function of $u = u(x)$ || $$\frac{\partial^2 \mathbf{x}^\top\mathbf{A}\mathbf{x}}{\partial\mathbf{x} \partial\mathbf{x}^\top} =$$ || colspan=2|$$\mathbf{A} + \mathbf{A}^\top$$
 * $A$ is not a function of $u = u(x)$
 * $u = u(x)$ is not a function of $u = u(x)$
 * $v = v(x)$ is not a function of $u = u(x)$
 * || $$\frac{\partial (\mathbf{x} \cdot \mathbf{x})}{\partial \mathbf{x}} = \frac{\partial \mathbf{x}^\top\mathbf{x}}{\partial \mathbf{x}} = \frac{\partial \left\Vert \mathbf{x} \right\Vert^2}{\partial \mathbf{x}} =$$ || $$2\mathbf{x}^\top $$ || $$2\mathbf{x}$$
 * $v = v(x)$ is not a function of $u = u(x)$,
 * $u = u(x)$ is not a function of $U = U(x)$,
 * $v = v(x)$ is not a function of $X^{T}$,
 * $$\frac{\partial (\mathbf{a} \cdot \mathbf{u})}{\partial \mathbf{x}} = \frac{\partial \mathbf{a}^\top\mathbf{u}}{\partial \mathbf{x}} =$$
 * $$\mathbf{a}^\top\frac{\partial \mathbf{u}}{\partial \mathbf{x}}$$
 * $$\frac{\partial \mathbf{u}}{\partial \mathbf{x}}\mathbf{a}$$
 * $X$, $a$ are not functions of $X$ || $$\frac{\partial \; \textbf{a}^\top\textbf{x}\textbf{x}^\top\textbf{b}}{\partial \; \textbf{x}} = $$ || $$\textbf{x}^\top\left(\textbf{a}\textbf{b}^\top + \textbf{b}\textbf{a}^\top\right)$$ || $$\left(\textbf{a}\textbf{b}^\top + \textbf{b}\textbf{a}^\top\right)\textbf{x}$$
 * $X$, $a$, $X$, $u = u(X)$, $u = u(X)$ are not functions of $v = v(X)$ || $$\frac{\partial \; (\textbf{A}\textbf{x} + \textbf{b})^\top \textbf{C} (\textbf{D}\textbf{x} + \textbf{e})}{\partial \; \textbf{x}} = $$ || $$(\textbf{D}\textbf{x} + \textbf{e})^\top \textbf{C}^\top \textbf{A} + (\textbf{A}\textbf{x} + \textbf{b})^\top \textbf{C} \textbf{D}$$ || $$ \textbf{D}^\top \textbf{C}^\top (\textbf{A}\textbf{x} + \textbf{b}) + \textbf{A}^\top\textbf{C}(\textbf{D}\textbf{x} + \textbf{e})$$
 * $u = u(X)$ is not a function of $v = v(X)$ || $$\frac{\partial \; \|\mathbf{x} - \mathbf{a}\|}{\partial \; \mathbf{x}} = $$ ||$$\frac{(\mathbf{x} - \mathbf{a})^\top}{\|\mathbf{x} - \mathbf{a}\|}$$||$$\frac{\mathbf{x} - \mathbf{a}}{\|\mathbf{x} - \mathbf{a}\|}$$
 * }
 * $u = u(X)$ is not a function of $u = u(X)$ || $$\frac{\partial \; \|\mathbf{x} - \mathbf{a}\|}{\partial \; \mathbf{x}} = $$ ||$$\frac{(\mathbf{x} - \mathbf{a})^\top}{\|\mathbf{x} - \mathbf{a}\|}$$||$$\frac{\mathbf{x} - \mathbf{a}}{\|\mathbf{x} - \mathbf{a}\|}$$
 * }
 * }

Vector-by-scalar identities

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="150" | Condition ! scope="col" width="100" | Expression ! scope="col" width="100" | Numerator layout, i.e. by $U = U(X)$, result is column vector ! scope="col" width="100" | Denominator layout, i.e. by $X$, result is row vector $a$ || $$\frac{\partial a\mathbf{u}}{\partial x} =$$ $b$ || $$\frac{\partial \mathbf{A}\mathbf{u}}{\partial x} =$$ || $$\mathbf{A}\frac{\partial \mathbf{u}}{\partial x}$$ || $$\frac{\partial \mathbf{u}}{\partial x}\mathbf{A}^\top$$
 * + Identities: vector-by-scalar $$\frac{\partial \mathbf{y}}{\partial x}$$
 * $X$ is not a function of $a$ || $$\frac{\partial \mathbf{a}}{\partial x} =$$ ||colspan=2| $$\mathbf{0}$$
 * $b$ is not a function of $X$,
 * $a$ is not a function of $b$,
 * $C$ is not a function of $X$,
 * colspan=2|$$a\frac{\partial \mathbf{u}}{\partial x}$$
 * $a$ is not a function of x,
 * $b$ is not a function of x,
 * $C$ || $$\frac{\partial \mathbf{u}^\top}{\partial x} =$$
 * colspan=2|$$\left(\frac{\partial \mathbf{u}}{\partial x}\right)^\top$$
 * $X$, $U = U(X)$ || $$\frac{\partial (\mathbf{u} + \mathbf{v})}{\partial x} =$$
 * colspan=2|$$\frac{\partial \mathbf{u}}{\partial x} + \frac{\partial \mathbf{v}}{\partial x}$$
 * $V = V(X)$, $a$ || $$\frac{\partial (\mathbf{u}^\top \times \mathbf{v})}{\partial x} =$$
 * $$\left(\frac{\partial \mathbf{u}}{\partial x}\right)^\top \times \mathbf{v} + \mathbf{u}^\top \times \frac{\partial \mathbf{v}}{\partial x}$$
 * $$\frac{\partial \mathbf{u}}{\partial x} \times \mathbf{v} + \mathbf{u}^\top \times \left(\frac{\partial \mathbf{v}}{\partial x}\right)^\top$$
 * rowspan=2|$X$ || rowspan=2|$$\frac{\partial \mathbf{g}(\mathbf{u})}{\partial x} =$$ || $$\frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial x}$$ || $$\frac{\partial \mathbf{u}}{\partial x} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}}$$
 * colspan=2|Assumes consistent matrix layout; see below.
 * rowspan=2| $U = U(X)$ || rowspan=2|$$\frac{\partial \mathbf{f}(\mathbf{g}(\mathbf{u}))}{\partial x} =$$ || $$\frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial x}$$ || $$\frac{\partial \mathbf{u}}{\partial x} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}}$$
 * colspan=2|Assumes consistent matrix layout; see below.
 * $g(X)$, $e^{X}$ || $$\frac{\partial (\mathbf{U} \times \mathbf{v})}{\partial x} =$$
 * $$\frac{\partial \mathbf{U}}{\partial x} \times \mathbf{v} + \mathbf{U} \times \frac{\partial \mathbf{v}}{\partial x}$$
 * $$\mathbf{v}^\top \times \left(\frac{\partial \mathbf{U}}{\partial x}\right) + \frac{\partial \mathbf{v}}{\partial x} \times \mathbf{U}^\top$$
 * }
 * rowspan=2| $sin(X)$ || rowspan=2|$$\frac{\partial \mathbf{f}(\mathbf{g}(\mathbf{u}))}{\partial x} =$$ || $$\frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{u}}{\partial x}$$ || $$\frac{\partial \mathbf{u}}{\partial x} \frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}} \frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}}$$
 * colspan=2|Assumes consistent matrix layout; see below.
 * $cos(X)$, $ln(X)$ || $$\frac{\partial (\mathbf{U} \times \mathbf{v})}{\partial x} =$$
 * $$\frac{\partial \mathbf{U}}{\partial x} \times \mathbf{v} + \mathbf{U} \times \frac{\partial \mathbf{v}}{\partial x}$$
 * $$\mathbf{v}^\top \times \left(\frac{\partial \mathbf{U}}{\partial x}\right) + \frac{\partial \mathbf{v}}{\partial x} \times \mathbf{U}^\top$$
 * }
 * $$\mathbf{v}^\top \times \left(\frac{\partial \mathbf{U}}{\partial x}\right) + \frac{\partial \mathbf{v}}{\partial x} \times \mathbf{U}^\top$$
 * }

NOTE: The formulas involving the vector-by-vector derivatives $$\frac{\partial \mathbf{g}(\mathbf{u})}{\partial \mathbf{u}}$$ and $$\frac{\partial \mathbf{f}(\mathbf{g})}{\partial \mathbf{g}}$$ (whose outputs are matrices) assume the matrices are laid out consistent with the vector layout, i.e. numerator-layout matrix when numerator-layout vector and vice versa; otherwise, transpose the vector-by-vector derivatives.

Scalar-by-matrix identities
Note that exact equivalents of the scalar product rule and chain rule do not exist when applied to matrix-valued functions of matrices. However, the product rule of this sort does apply to the differential form (see below), and this is the way to derive many of the identities below involving the trace function, combined with the fact that the trace function allows transposing and cyclic permutation, i.e.:
 * $$\begin{align}

\operatorname{tr}(\mathbf{A}) &= \operatorname{tr}\left(\mathbf{A^\top}\right) \\ \operatorname{tr}(\mathbf{ABCD}) &= \operatorname{tr}(\mathbf{BCDA}) = \operatorname{tr}(\mathbf{CDAB}) = \operatorname{tr}(\mathbf{DABC}) \end{align}$$

For example, to compute $$\frac{\partial \operatorname{tr}(\mathbf{AXBX^\top C})}{\partial \mathbf{X}}:$$ $$\begin{align} d\operatorname{tr}(\mathbf{AXBX^\top C}) &= d\operatorname{tr}\left(\mathbf{CAXBX^\top}\right) = \operatorname{tr}\left(d\left(\mathbf{CAXBX^\top}\right)\right) \\[1ex] &= \operatorname{tr}\left(\mathbf{CAX} d(\mathbf{BX^\top}\right) + d\left(\mathbf{CAX})\mathbf{BX^\top}\right) \\[1ex] &= \operatorname{tr}\left(\mathbf{CAX} d\left(\mathbf{BX^\top}\right)\right) + \operatorname{tr}\left(d(\mathbf{CAX})\mathbf{BX^\top}\right) \\[1ex] &= \operatorname{tr}\left(\mathbf{CAXB} d\left(\mathbf{X^\top}\right)\right) + \operatorname{tr}\left(\mathbf{CA}(d\mathbf{X})\mathbf{BX^\top}\right) \\[1ex] &= \operatorname{tr}\left(\mathbf{CAXB} (d\mathbf{X})^\top\right) + \operatorname{tr}(\mathbf{CA}\left(d\mathbf{X})\mathbf{BX^\top}\right) \\[1ex] &= \operatorname{tr}\left(\left(\mathbf{CAXB} (d\mathbf{X})^\top\right)^\top\right) + \operatorname{tr}\left(\mathbf{CA}(d\mathbf{X})\mathbf{BX^\top}\right) \\[1ex] &= \operatorname{tr}\left((d\mathbf{X})\mathbf{B^\top X^\top A^\top C^\top}\right) + \operatorname{tr}\left(\mathbf{CA}(d\mathbf{X})\mathbf{BX^\top}\right) \\[1ex] &= \operatorname{tr}\left(\mathbf{B^\top X^\top A^\top C^\top}(d\mathbf{X})\right) + \operatorname{tr}\left(\mathbf{BX^\top}\mathbf{CA}(d\mathbf{X})\right) \\[1ex] &= \operatorname{tr}\left(\left(\mathbf{B^\top X^\top A^\top C^\top} + \mathbf{BX^\top}\mathbf{CA}\right)d\mathbf{X}\right) \\[1ex] &= \operatorname{tr}\left(\left(\mathbf{C A X B} + \mathbf{A^\top C^\top X B^\top }\right)^\top d\mathbf{X}\right) \end{align}$$

Therefore,

 * $$\frac{\partial \operatorname{tr}\left(\mathbf{AXBX^\top C}\right)}{\partial \mathbf{X}} = \mathbf{B^\top X^\top A^\top C^\top} + \mathbf{B X^\top C A}.$$ (numerator layout)
 * $$\frac{\partial \operatorname{tr}\left(\mathbf{AXBX^\top C}\right)}{\partial \mathbf{X}} = \mathbf{C A X B} + \mathbf{A^\top C^\top X B^\top}.$$ (denominator layout)

(For the last step, see the Conversion from differential to derivative form section.)

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="175" | Condition ! scope="col" width="10" | Expression ! scope="col" width="100" | Numerator layout, i.e. by $g(x)$ ! scope="col" width="100" | Denominator layout, i.e. by $g ′ (x)$
 * + Identities: scalar-by-matrix $$\frac{\partial y}{\partial \mathbf{X}}$$
 * $g ′ (X)$ is not a function of $A$ || $$\frac{\partial a}{\partial \mathbf{X}} =$$
 * $$\mathbf{0}^\top$$ ||$$\mathbf{0}$$
 * $X$ is not a function of $A$, $X$ || $$\frac{\partial au}{\partial \mathbf{X}} =$$
 * colspan=2|$$a\frac{\partial u}{\partial \mathbf{X}}$$
 * $A$, $X$ || $$\frac{\partial (u+v)}{\partial \mathbf{X}} =$$
 * colspan=2|$$\frac{\partial u}{\partial \mathbf{X}} + \frac{\partial v}{\partial \mathbf{X}} $$
 * $A$, $X$ || $$\frac{\partial uv}{\partial \mathbf{X}} =$$
 * colspan=2|$$u\frac{\partial v}{\partial \mathbf{X}} + v\frac{\partial u}{\partial \mathbf{X}} $$
 * $A$ || $$\frac{\partial g(u)}{\partial \mathbf{X}} =$$
 * colspan=2|$$\frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf{X}} $$
 * $B$ || $$\frac{\partial f(g(u))}{\partial \mathbf{X}} =$$
 * colspan=2|$$\frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf{X}} $$
 * rowspan=2| $X$ || rowspan=2|   $$\frac{\partial g(\mathbf{U})}{\partial X_{ij}} =$$
 * $$\operatorname{tr}\left( \frac{\partial g(\mathbf{U})}{\partial \mathbf{U}} \frac{\partial \mathbf{U}}{\partial X_{ij}}\right)$$
 * $$\operatorname{tr}\left( \left(\frac{\partial g(\mathbf{U})}{\partial \mathbf{U}}\right)^\top \frac{\partial \mathbf{U}}{\partial X_{ij}}\right)$$
 * colspan=2|Both forms assume numerator layout for $$\frac{\partial \mathbf{U}}{\partial X_{ij}},$$
 * $A$ || $$\frac{\partial f(g(u))}{\partial \mathbf{X}} =$$
 * colspan=2|$$\frac{\partial f(g)}{\partial g} \frac{\partial g(u)}{\partial u} \frac{\partial u}{\partial \mathbf{X}} $$
 * rowspan=2| $B$ || rowspan=2|   $$\frac{\partial g(\mathbf{U})}{\partial X_{ij}} =$$
 * $$\operatorname{tr}\left( \frac{\partial g(\mathbf{U})}{\partial \mathbf{U}} \frac{\partial \mathbf{U}}{\partial X_{ij}}\right)$$
 * $$\operatorname{tr}\left( \left(\frac{\partial g(\mathbf{U})}{\partial \mathbf{U}}\right)^\top \frac{\partial \mathbf{U}}{\partial X_{ij}}\right)$$
 * colspan=2|Both forms assume numerator layout for $$\frac{\partial \mathbf{U}}{\partial X_{ij}},$$
 * colspan=2|Both forms assume numerator layout for $$\frac{\partial \mathbf{U}}{\partial X_{ij}},$$
 * colspan=2|Both forms assume numerator layout for $$\frac{\partial \mathbf{U}}{\partial X_{ij}},$$

i.e. mixed layout if denominator layout for $C$ is being used. $X$ || $$\frac{\partial \operatorname{tr}(a\mathbf{U})}{\partial \mathbf{X}} =$$ || colspan=2|$$a\frac{\partial \operatorname{tr}(\mathbf{U})}{\partial \mathbf{X}}$$ $n$ is a positive integer ||   $$\frac{\partial \operatorname{tr}\left(\mathbf{A}\mathbf{X}^n\right)}{\partial \mathbf{X}} =$$ || $$\sum_{i=0}^{n-1} \mathbf{X}^i\mathbf{A}\mathbf{X}^{n-i-1}$$ || $$\sum_{i=0}^{n-1} \left(\mathbf{X}^i\mathbf{A}\mathbf{X}^{n-i-1}\right)^\top$$ || $$\mathbf{X}^{-1}$$ ||$$\left(\mathbf{X}^{-1}\right)^\top$$ $A$ is square and invertible || $$\frac{\partial \left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right|}{\partial \mathbf{X}} =$$ || $$2\left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right|\mathbf{X}^{-1} = 2\left|\mathbf{X^\top}\right||\mathbf{A}||\mathbf{X}|\mathbf{X}^{-1}$$ || $$2\left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right|\left(\mathbf{X}^{-1}\right)^\top$$ $X$ is non-square, $a$ is symmetric || $$\frac{\partial \left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right|}{\partial \mathbf{X}} =$$ || $$2\left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right|\left(\mathbf{X^\top A^\top X}\right)^{-1}\mathbf{X^\top A^\top}$$ || $$2\left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right|\mathbf{AX}\left(\mathbf{X^\top AX}\right)^{-1}$$ $X$ is non-square, $a$ is non-symmetric || $$\frac{\partial |\mathbf{X^\top}\mathbf{A}\mathbf{X}|}{\partial \mathbf{X}} =$$ \left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right| \Big(&\left(\mathbf{X^\top AX}\right)^{-1}\mathbf{X^\top A} + {} \\        &\left(\mathbf{X^\top A^\top X}\right)^{-1}\mathbf{X^\top A^\top}\Big) \end{align}$$ \left|\mathbf{X^\top}\mathbf{A}\mathbf{X}\right| \Big(&\mathbf{AX}\left(\mathbf{X^\top AX}\right)^{-1} + {} \\        &\mathbf{A^\top X}\left(\mathbf{X^\top A^\top X}\right)^{-1}\Big) \end{align}$$
 * - style="border-top: 3px solid;"
 * $A$ and $B$ are not functions of $X$ ||$$\frac{\partial \mathbf{a}^\top\mathbf{X}\mathbf{b}}{\partial \mathbf{X}} =$$
 * $$\mathbf{b}\mathbf{a}^\top$$||$$\mathbf{a}\mathbf{b}^\top$$
 * $A$ and $X$ are not functions of $X$ ||$$\frac{\partial \mathbf{a}^\top \mathbf{X}^\top \mathbf{b}}{\partial \mathbf{X}} =$$
 * $$\mathbf{a}\mathbf{b}^\top$$||$$\mathbf{b}\mathbf{a}^\top$$
 * $A$, $X$ and $X$ are not functions of $A$ ||$$\frac{\partial (\mathbf{X}\mathbf{a} + \mathbf{b})^\top \mathbf{C}(\mathbf{X} \mathbf{a} + \mathbf{b})}{\partial \mathbf{X}} =$$
 * $$\left(\left(\mathbf{C} + \mathbf{C}^\top\right)(\mathbf{X} \mathbf{a} + \mathbf{b})\mathbf{a}^\top\right)^\top$$||$$\left(\mathbf{C} + \mathbf{C}^\top\right)(\mathbf{X} \mathbf{a} + \mathbf{b})\mathbf{a}^\top$$
 * $A$, $X$ and $X$ are not functions of $A$ ||$$\frac{\partial (\mathbf{X}\mathbf{a})^\top \mathbf{C}(\mathbf{X}\mathbf{b})}{\partial \mathbf{X}} =$$
 * $$\left(\mathbf{C}\mathbf{X}\mathbf{b}\mathbf{a}^\top + \mathbf{C}^\top\mathbf{X}\mathbf{a}\mathbf{b}^\top\right)^\top$$||$$\mathbf{C}\mathbf{X}\mathbf{b}\mathbf{a}^\top + \mathbf{C}^\top\mathbf{X}\mathbf{a}\mathbf{b}^\top$$
 * - style="border-top: 3px solid;"
 * || $$\frac{\partial \operatorname{tr}(\mathbf{X})}{\partial \mathbf{X}} =$$ || colspan=2|$$\mathbf{I}$$
 * $Y$, $U = U(x)$ || $$\frac{\partial \operatorname{tr}(\mathbf{U} + \mathbf{V})}{\partial \mathbf{X}} =$$ || colspan=2|$$\frac{\partial \operatorname{tr}(\mathbf{U})}{\partial \mathbf{X}} + \frac{\partial \operatorname{tr}(\mathbf{V})}{\partial \mathbf{X}}$$
 * $A$ is not a function of $B$,
 * || $$\frac{\partial \operatorname{tr}(\mathbf{X})}{\partial \mathbf{X}} =$$ || colspan=2|$$\mathbf{I}$$
 * $U = U(x)$, $U = U(x)$ || $$\frac{\partial \operatorname{tr}(\mathbf{U} + \mathbf{V})}{\partial \mathbf{X}} =$$ || colspan=2|$$\frac{\partial \operatorname{tr}(\mathbf{U})}{\partial \mathbf{X}} + \frac{\partial \operatorname{tr}(\mathbf{V})}{\partial \mathbf{X}}$$
 * $V = V(x)$ is not a function of $U = U(x)$,
 * $V = V(x)$ is not a function of $U = U(x)$,
 * $V = V(x)$ is not a function of $U = U(x)$,
 * $V = V(x)$ is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. $U = U(x)$, $U = U(x,y)$, $A$, $x$, etc. using a Taylor series); $g(X)$ is the equivalent scalar function, $e^{X}$ is its derivative, and $sin(X)$ is the corresponding matrix function || $$\frac{\partial \operatorname{tr}(\mathbf{g(X)})}{\partial \mathbf{X}} =$$ || $$\mathbf{g}'(\mathbf{X})$$ || $$\left(\mathbf{g}'(\mathbf{X})\right)^\top$$
 * $cos(X)$ is not a function of $ln(X)$ ||   $$\frac{\partial \operatorname{tr}(\mathbf{AX})}{\partial \mathbf{X}} = \frac{\partial \operatorname{tr}(\mathbf{XA})}{\partial \mathbf{X}} =$$ || $$\mathbf{A}$$ || $$\mathbf{A}^\top$$
 * $g(x)$ is not a function of $g ′ (x)$ ||   $$\frac{\partial \operatorname{tr}\left(\mathbf{AX^\top}\right)}{\partial \mathbf{X}} = \frac{\partial \operatorname{tr}\left(\mathbf{X^\top A}\right)}{\partial \mathbf{X}} =$$ || $$\mathbf{A}^\top$$ || $$\mathbf{A}$$
 * $g ′ (X)$ is not a function of $A$ ||   $$\frac{\partial \operatorname{tr}\left(\mathbf{X^\top AX}\right)}{\partial \mathbf{X}} =$$ || $$\mathbf{X}^\top\left(\mathbf{A} + \mathbf{A}^\top\right)$$ || $$\left(\mathbf{A} + \mathbf{A}^\top\right)\mathbf{X}$$
 * $x$ is not a function of $u = u(x)$ ||   $$\frac{\partial \operatorname{tr}(\mathbf{X^{-1}A})}{\partial \mathbf{X}} =$$ || $$-\mathbf{X}^{-1}\mathbf{A}\mathbf{X}^{-1}$$ || $$-\left(\mathbf{X}^{-1}\right)^\top\mathbf{A}^\top\left(\mathbf{X}^{-1}\right)^\top$$
 * $u = u(x)$, $v = v(x)$ are not functions of $Y$ || $$\frac{\partial \operatorname{tr}(\mathbf{AXB})}{\partial \mathbf{X}} = \frac{\partial \operatorname{tr}(\mathbf{BAX})}{\partial \mathbf{X}} =$$ || $$\mathbf{BA}$$ || $$\mathbf{A^\top B^\top}$$
 * $X$, $Y$, $X^{T}$ are not functions of $Y$ || $$\frac{\partial \operatorname{tr}\left(\mathbf{AXBX^\top C}\right)}{\partial \mathbf{X}} = $$ || $$\mathbf{BX^\top CA} + \mathbf{B^\top X^\top A^\top C^\top}$$ || $$\mathbf{A^\top C^\top XB^\top} + \mathbf{CAXB}$$
 * $n$ is a positive integer ||   $$\frac{\partial \operatorname{tr} \left(\mathbf{X}^n\right)}{\partial \mathbf{X}} =$$ || $$n\mathbf{X}^{n-1}$$ || $$n\left(\mathbf{X}^{n-1}\right)^\top$$
 * $X$ is not a function of $U = U(x)$,
 * $U = U(x)$ is not a function of $U = U(x)$ ||   $$\frac{\partial \operatorname{tr}(\mathbf{X^{-1}A})}{\partial \mathbf{X}} =$$ || $$-\mathbf{X}^{-1}\mathbf{A}\mathbf{X}^{-1}$$ || $$-\left(\mathbf{X}^{-1}\right)^\top\mathbf{A}^\top\left(\mathbf{X}^{-1}\right)^\top$$
 * $U = U(x)$, $A$ are not functions of $x$ || $$\frac{\partial \operatorname{tr}(\mathbf{AXB})}{\partial \mathbf{X}} = \frac{\partial \operatorname{tr}(\mathbf{BAX})}{\partial \mathbf{X}} =$$ || $$\mathbf{BA}$$ || $$\mathbf{A^\top B^\top}$$
 * $g(X)$, $e^{X}$, $sin(X)$ are not functions of $cos(X)$ || $$\frac{\partial \operatorname{tr}\left(\mathbf{AXBX^\top C}\right)}{\partial \mathbf{X}} = $$ || $$\mathbf{BX^\top CA} + \mathbf{B^\top X^\top A^\top C^\top}$$ || $$\mathbf{A^\top C^\top XB^\top} + \mathbf{CAXB}$$
 * $n$ is a positive integer ||   $$\frac{\partial \operatorname{tr} \left(\mathbf{X}^n\right)}{\partial \mathbf{X}} =$$ || $$n\mathbf{X}^{n-1}$$ || $$n\left(\mathbf{X}^{n-1}\right)^\top$$
 * $ln(X)$ is not a function of $g(x)$,
 * $n$ is a positive integer ||   $$\frac{\partial \operatorname{tr} \left(\mathbf{X}^n\right)}{\partial \mathbf{X}} =$$ || $$n\mathbf{X}^{n-1}$$ || $$n\left(\mathbf{X}^{n-1}\right)^\top$$
 * $g ′ (x)$ is not a function of $g ′ (X)$,
 * $A$ is not a function of $x$,
 * $a$ is not a function of $X$,
 * ||   $$\frac{\partial \operatorname{tr}\left(e^\mathbf{X}\right)}{\partial \mathbf{X}} =$$ || $$e^\mathbf{X}$$ || $$\left(e^\mathbf{X}\right)^\top$$
 * ||   $$\frac{\partial \operatorname{tr}(\sin(\mathbf{X}))}{\partial \mathbf{X}} =$$ || $$\cos(\mathbf{X})$$ || $$(\cos(\mathbf{X}))^\top$$
 * - style="border-top: 3px solid;"
 * ||   $$\frac{\partial |\mathbf{X}|}{\partial \mathbf{X}} =$$ || $$\operatorname{cofactor}(X)^\top = |\mathbf{X}|\mathbf{X}^{-1}$$ ||$$\operatorname{cofactor}(X) = |\mathbf{X}|\left(\mathbf{X}^{-1}\right)^\top$$
 * $X$ is not a function of $f$ || $$\frac{\partial \ln |a\mathbf{X}|}{\partial \mathbf{X}} =$$
 * ||   $$\frac{\partial |\mathbf{X}|}{\partial \mathbf{X}} =$$ || $$\operatorname{cofactor}(X)^\top = |\mathbf{X}|\mathbf{X}^{-1}$$ ||$$\operatorname{cofactor}(X) = |\mathbf{X}|\left(\mathbf{X}^{-1}\right)^\top$$
 * $X$ is not a function of $Q$ || $$\frac{\partial \ln |a\mathbf{X}|}{\partial \mathbf{X}} =$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭ || $$\frac{\partial \ln |a\mathbf{X}|}{\partial \mathbf{X}} =$$
 * ᙭᙭᙭, ᙭᙭᙭ are not functions of ᙭᙭᙭ ||    $$\frac{\partial |\mathbf{AXB}|}{\partial \mathbf{X}} =$$ || $$|\mathbf{AXB}|\mathbf{X}^{-1}$$ ||$$|\mathbf{AXB}|\left(\mathbf{X}^{-1}\right)^\top$$
 * $n$ is a positive integer ||   $$\frac{\partial \left|\mathbf{X}^n\right|}{\partial \mathbf{X}} =$$ || $$n\left|\mathbf{X}^n\right|\mathbf{X}^{-1}$$ ||$$n\left|\mathbf{X}^n\right|\left(\mathbf{X}^{-1}\right)^\top$$
 * (see pseudo-inverse) ||     $$\frac{\partial \ln \left|\mathbf{X}^\top\mathbf{X}\right|}{\partial \mathbf{X}} =$$ || $$2\mathbf{X}^{+}$$ ||$$2\left(\mathbf{X}^{+}\right)^\top$$
 * (see pseudo-inverse) ||    $$\frac{\partial \ln \left|\mathbf{X}^\top\mathbf{X}\right|}{\partial \mathbf{X}^{+}} =$$ || $$-2\mathbf{X}$$ || $$-2\mathbf{X}^\top$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * (see pseudo-inverse) ||     $$\frac{\partial \ln \left|\mathbf{X}^\top\mathbf{X}\right|}{\partial \mathbf{X}} =$$ || $$2\mathbf{X}^{+}$$ ||$$2\left(\mathbf{X}^{+}\right)^\top$$
 * (see pseudo-inverse) ||    $$\frac{\partial \ln \left|\mathbf{X}^\top\mathbf{X}\right|}{\partial \mathbf{X}^{+}} =$$ || $$-2\mathbf{X}$$ || $$-2\mathbf{X}^\top$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * ᙭᙭᙭ is not a function of ᙭᙭᙭,
 * $$\begin{align}
 * $$\begin{align}
 * }

Matrix-by-scalar identities

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="175" | Condition ! scope="col" width="100" | Expression ! scope="col" width="100" | Numerator layout, i.e. by ᙭᙭᙭ ᙭᙭᙭ || $$\frac{\partial \mathbf{AUB}}{\partial x} =$$ || $$\mathbf{A}\frac{\partial \mathbf{U}}{\partial x}\mathbf{B}$$
 * + Identities: matrix-by-scalar $$\frac{\partial \mathbf{Y}}{\partial x}$$
 * ᙭᙭᙭ || $$\frac{\partial a\mathbf{U}}{\partial x} =$$ || $$a\frac{\partial \mathbf{U}}{\partial x}$$
 * ᙭᙭᙭, ᙭᙭᙭ are not functions of x,
 * ᙭᙭᙭, ᙭᙭᙭ are not functions of x,
 * ᙭᙭᙭, ᙭᙭᙭ are not functions of x,
 * ᙭᙭᙭, ᙭᙭᙭ || $$\frac{\partial (\mathbf{U}+\mathbf{V})}{\partial x} =$$ || $$\frac{\partial \mathbf{U}}{\partial x} + \frac{\partial \mathbf{V}}{\partial x}$$
 * ᙭᙭᙭, ᙭᙭᙭ || $$\frac{\partial (\mathbf{U}\mathbf{V})}{\partial x} =$$ || $$\mathbf{U}\frac{\partial \mathbf{V}}{\partial x} + \frac{\partial \mathbf{U}}{\partial x}\mathbf{V}$$
 * ᙭᙭᙭, ᙭᙭᙭ || $$\frac{\partial (\mathbf{U} \otimes \mathbf{V})}{\partial x} =$$ || $$\mathbf{U} \otimes \frac{\partial \mathbf{V}}{\partial x} + \frac{\partial \mathbf{U}}{\partial x} \otimes \mathbf{V}$$
 * ᙭᙭᙭, ᙭᙭᙭ || $$\frac{\partial (\mathbf{U} \circ \mathbf{V})}{\partial x} =$$ || $$\mathbf{U} \circ \frac{\partial \mathbf{V}}{\partial x} + \frac{\partial \mathbf{U}}{\partial x} \circ \mathbf{V}$$
 * ᙭᙭᙭ || $$\frac{\partial \mathbf{U}^{-1}}{\partial x} =$$ || $$-\mathbf{U}^{-1} \frac{\partial \mathbf{U}}{\partial x}\mathbf{U}^{-1}$$
 * ᙭᙭᙭ || $$\frac{\partial^2 \mathbf{U}^{-1}}{\partial x \partial y} =$$ || $$\mathbf{U}^{-1}\left(\frac{\partial \mathbf{U}}{\partial x}\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial y} - \frac{\partial^2 \mathbf{U}}{\partial x \partial y} + \frac{\partial \mathbf{U}}{\partial y}\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial x}\right)\mathbf{U}^{-1}$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭, ᙭᙭᙭ is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, etc.); ᙭᙭᙭ is the equivalent scalar function, ᙭᙭᙭ is its derivative, and ᙭᙭᙭ is the corresponding matrix function || $$\frac{\partial \, \mathbf{g}(x\mathbf{A})}{\partial x} =$$ ||$$\mathbf{A}\mathbf{g}'(x\mathbf{A}) = \mathbf{g}'(x\mathbf{A})\mathbf{A}$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭ || $$\frac{\partial e^{x\mathbf{A}}}{\partial x} =$$ || $$\mathbf{A} e^{x\mathbf{A}} = e^{x\mathbf{A}} \mathbf{A}$$
 * }
 * ᙭᙭᙭ || $$\frac{\partial \mathbf{U}^{-1}}{\partial x} =$$ || $$-\mathbf{U}^{-1} \frac{\partial \mathbf{U}}{\partial x}\mathbf{U}^{-1}$$
 * ᙭᙭᙭ || $$\frac{\partial^2 \mathbf{U}^{-1}}{\partial x \partial y} =$$ || $$\mathbf{U}^{-1}\left(\frac{\partial \mathbf{U}}{\partial x}\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial y} - \frac{\partial^2 \mathbf{U}}{\partial x \partial y} + \frac{\partial \mathbf{U}}{\partial y}\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial x}\right)\mathbf{U}^{-1}$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭, ᙭᙭᙭ is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, etc.); ᙭᙭᙭ is the equivalent scalar function, ᙭᙭᙭ is its derivative, and ᙭᙭᙭ is the corresponding matrix function || $$\frac{\partial \, \mathbf{g}(x\mathbf{A})}{\partial x} =$$ ||$$\mathbf{A}\mathbf{g}'(x\mathbf{A}) = \mathbf{g}'(x\mathbf{A})\mathbf{A}$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭ || $$\frac{\partial e^{x\mathbf{A}}}{\partial x} =$$ || $$\mathbf{A} e^{x\mathbf{A}} = e^{x\mathbf{A}} \mathbf{A}$$
 * }
 * ᙭᙭᙭ is not a function of ᙭᙭᙭ || $$\frac{\partial e^{x\mathbf{A}}}{\partial x} =$$ || $$\mathbf{A} e^{x\mathbf{A}} = e^{x\mathbf{A}} \mathbf{A}$$
 * }
 * }

With vectors involved

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="150" | Condition ! scope="col" width="10" | Expression ! scope="col" width="150" | Any layout (assumes dot product ignores row vs. column layout)
 * + Identities: scalar-by-scalar, with vectors involved
 * ᙭᙭᙭ || $$\frac{\partial g(\mathbf{u})}{\partial x} =$$ || $$\frac{\partial g(\mathbf{u})}{\partial \mathbf{u}} \cdot \frac{\partial \mathbf{u}}{\partial x}$$
 * ᙭᙭᙭, ᙭᙭᙭ || $$\frac{\partial (\mathbf{u} \cdot \mathbf{v})}{\partial x} =$$
 * $$\mathbf{u} \cdot \frac{\partial \mathbf{v}}{\partial x} + \frac{\partial \mathbf{u}}{\partial x} \cdot \mathbf{v} $$
 * }
 * $$\mathbf{u} \cdot \frac{\partial \mathbf{v}}{\partial x} + \frac{\partial \mathbf{u}}{\partial x} \cdot \mathbf{v} $$
 * }

With matrices involved

 * {|class="wikitable" style="text-align: center;"

! scope="col" width="175" | Condition ! scope="col" width="100" | Expression ! scope="col" width="100" | Consistent numerator layout, i.e. by ᙭᙭᙭ and ᙭᙭᙭ ! scope="col" width="100" | Mixed layout, i.e. by ᙭᙭᙭ and ᙭᙭᙭ \operatorname{tr}\left(\mathbf{U}^{-1}\frac{\partial^2 \mathbf{U}}{\partial x^2}\right) + \operatorname{tr}^2\left(\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial x}\right) - \operatorname{tr}\left(\left(\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial x}\right)^2\right) \right]$$
 * +Identities: scalar-by-scalar, with matrices involved
 * ᙭᙭᙭ || $$\frac{\partial |\mathbf{U}|}{\partial x} =$$ || colspan=2|$$|\mathbf{U}|\operatorname{tr}\left(\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial x}\right)$$
 * ᙭᙭᙭ || $$\frac{\partial \ln|\mathbf{U}|}{\partial x} =$$ || colspan=2|$$\operatorname{tr}\left(\mathbf{U}^{-1}\frac{\partial \mathbf{U}}{\partial x}\right)$$
 * ᙭᙭᙭ || $$\frac{\partial^2 |\mathbf{U}|}{\partial x^2} =$$
 * colspan=2 | $$\left|\mathbf{U}\right| \left[
 * ᙭᙭᙭ || $$\frac{\partial^2 |\mathbf{U}|}{\partial x^2} =$$
 * colspan=2 | $$\left|\mathbf{U}\right| \left[
 * colspan=2 | $$\left|\mathbf{U}\right| \left[
 * $$\frac{\partial g(\mathbf{U})}{\partial x} =$$
 * $$\operatorname{tr}\left( \frac{\partial g(\mathbf{U})}{\partial \mathbf{U}} \frac{\partial \mathbf{U}}{\partial x}\right)$$
 * $$\operatorname{tr}\left( \left(\frac{\partial g(\mathbf{U})}{\partial \mathbf{U}}\right)^\top \frac{\partial \mathbf{U}}{\partial x}\right)$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭, ᙭᙭᙭ is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, etc.); ᙭᙭᙭ is the equivalent scalar function, ᙭᙭᙭ is its derivative, and ᙭᙭᙭ is the corresponding matrix function. || $$\frac{\partial \operatorname{tr}(\mathbf{g}(x\mathbf{A}))}{\partial x} =$$ || colspan=2|$$\operatorname{tr}\left(\mathbf{A}\mathbf{g}'(x\mathbf{A})\right)$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭ || $$\frac{\partial \operatorname{tr}\left(e^{x\mathbf{A}}\right)}{\partial x} =$$ || colspan=2|$$\operatorname{tr}\left(\mathbf{A}e^{x\mathbf{A}}\right)$$
 * }
 * ᙭᙭᙭ is not a function of ᙭᙭᙭, ᙭᙭᙭ is any polynomial with scalar coefficients, or any matrix function defined by an infinite polynomial series (e.g. ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, ᙭᙭᙭, etc.); ᙭᙭᙭ is the equivalent scalar function, ᙭᙭᙭ is its derivative, and ᙭᙭᙭ is the corresponding matrix function. || $$\frac{\partial \operatorname{tr}(\mathbf{g}(x\mathbf{A}))}{\partial x} =$$ || colspan=2|$$\operatorname{tr}\left(\mathbf{A}\mathbf{g}'(x\mathbf{A})\right)$$
 * ᙭᙭᙭ is not a function of ᙭᙭᙭ || $$\frac{\partial \operatorname{tr}\left(e^{x\mathbf{A}}\right)}{\partial x} =$$ || colspan=2|$$\operatorname{tr}\left(\mathbf{A}e^{x\mathbf{A}}\right)$$
 * }
 * }

Identities in differential form
It is often easier to work in differential form and then convert back to normal derivatives. This only works well using the numerator layout. In these rules, ᙭᙭᙭ is a scalar.

 * {|class="wikitable" style="text-align: center;"

! Expression !! Result (numerator layout)
 * + Differential identities: scalar involving matrix
 * $$d(\operatorname{tr}(\mathbf{X})) =$$ || $$\operatorname{tr}(d\mathbf{X})$$
 * $$d(|\mathbf{X}|) =$$ || $$|\mathbf{X}|\operatorname{tr}\left(\mathbf{X}^{-1}d\mathbf{X}\right) = \operatorname{tr}(\operatorname{adj}(\mathbf{X})d\mathbf{X})$$
 * $$d(\ln|\mathbf{X}|) =$$ || $$\operatorname{tr}\left(\mathbf{X}^{-1}d\mathbf{X}\right)$$
 * }
 * $$d(\ln|\mathbf{X}|) =$$ || $$\operatorname{tr}\left(\mathbf{X}^{-1}d\mathbf{X}\right)$$
 * }
 * }

 * {|class="wikitable" style="text-align: center;"

! Condition !! Expression !! Result (numerator layout) $$\mathbf{P}_i \mathbf{P}_j = \delta_{ij} \mathbf{P}_i $$ ᙭᙭᙭ is differentiable at every eigenvalue $$\lambda_i$$ f'(\lambda_i) & \lambda_i = \lambda_j \\ \frac{f(\lambda_i) - f(\lambda_j)}{\lambda_i - \lambda_j} & \lambda_i \neq \lambda_j \end{cases} $$ In the last row, $$\delta_{ij}$$ is the Kronecker delta and $$(\mathbf{P}_k)_{ij} = (\mathbf{Q})_{ik} (\mathbf{Q}^{-1})_{kj}$$ is the set of orthogonal projection operators that project onto the $n$-th eigenvector of ᙭᙭᙭. ᙭᙭᙭ is the matrix of eigenvectors of $$\mathbf{X} = \mathbf{Q} \boldsymbol{\Lambda} \mathbf{Q}^{-1}$$, and $$(\boldsymbol{\Lambda})_{ii} = \lambda_i$$ are the eigenvalues. The matrix function $$f(\mathbf{X})$$ is defined in terms of the scalar function $$f(x)$$ for diagonalizable matrices by $f(\mathbf{X}) = \sum_i f(\lambda_i) \mathbf{P}_i $ where $\mathbf{X} = \sum_i \lambda_i \mathbf{P}_i$  with $\mathbf{P}_i \mathbf{P}_j = \delta_{ij} \mathbf{P}_i $.
 * + Differential identities: matrix
 * A is not a function of ᙭᙭᙭ || $$d(\mathbf{A}) =$$ || $$0$$
 * a is not a function of ᙭᙭᙭ || $$d(a\mathbf{X}) =$$ || $$a\,d\mathbf{X}$$
 * || $$d(\mathbf{X} + \mathbf{Y}) =$$ || $$d\mathbf{X} + d\mathbf{Y}$$
 * || $$d(\mathbf{X}\mathbf{Y}) =$$ || $$(d\mathbf{X})\mathbf{Y} + \mathbf{X}(d\mathbf{Y})$$
 * (Kronecker product) || $$d(\mathbf{X} \otimes \mathbf{Y}) =$$ || $$(d\mathbf{X})\otimes\mathbf{Y} + \mathbf{X}\otimes(d\mathbf{Y})$$
 * (Hadamard product) || $$d(\mathbf{X} \circ \mathbf{Y}) =$$ || $$(d\mathbf{X}) \circ \mathbf{Y} + \mathbf{X} \circ (d\mathbf{Y})$$
 * || $$d\left(\mathbf{X}^\top\right) =$$ || $$(d\mathbf{X})^\top$$
 * $$d\left(\mathbf{X}^{-1}\right) =$$
 * $$-\mathbf{X}^{-1}\left(d\mathbf{X}\right)\mathbf{X}^{-1}$$
 * (conjugate transpose) || $$d\left(\mathbf{X}^\mathrm{H}\right) =$$ || $$(d\mathbf{X})^\mathrm{H}$$
 * $k$ is a positive integer || $$d\left(\mathbf{X}^n\right) =$$ || $$\sum_{i=0}^{n-1} \mathbf{X}^i (d\mathbf{X})\mathbf{X}^{n-i-1}$$
 * $$d \left(e^\mathbf{X}\right) =$$
 * $$ \int_0^1 e^{a\mathbf{X}} (d\mathbf{X}) e^{(1-a)\mathbf{X}} \, da $$
 * $$d \left(\log{X}\right) =$$
 * $$ \int_0^\infty (\mathbf{X}+z \, \mathbf{I})^{-1} (d\mathbf{X}) (\mathbf{X}+z \, \mathbf{I})^{-1} \, dz $$
 * $$\mathbf{X} = \sum_i \lambda_i \mathbf{P}_i$$ is diagonalizable
 * $$d\left(\mathbf{X}^{-1}\right) =$$
 * $$-\mathbf{X}^{-1}\left(d\mathbf{X}\right)\mathbf{X}^{-1}$$
 * (conjugate transpose) || $$d\left(\mathbf{X}^\mathrm{H}\right) =$$ || $$(d\mathbf{X})^\mathrm{H}$$
 * ⇭⇭⇭ is a positive integer || $$d\left(\mathbf{X}^n\right) =$$ || $$\sum_{i=0}^{n-1} \mathbf{X}^i (d\mathbf{X})\mathbf{X}^{n-i-1}$$
 * $$d \left(e^\mathbf{X}\right) =$$
 * $$ \int_0^1 e^{a\mathbf{X}} (d\mathbf{X}) e^{(1-a)\mathbf{X}} \, da $$
 * $$d \left(\log{X}\right) =$$
 * $$ \int_0^\infty (\mathbf{X}+z \, \mathbf{I})^{-1} (d\mathbf{X}) (\mathbf{X}+z \, \mathbf{I})^{-1} \, dz $$
 * $$\mathbf{X} = \sum_i \lambda_i \mathbf{P}_i$$ is diagonalizable
 * $$ \int_0^1 e^{a\mathbf{X}} (d\mathbf{X}) e^{(1-a)\mathbf{X}} \, da $$
 * $$d \left(\log{X}\right) =$$
 * $$ \int_0^\infty (\mathbf{X}+z \, \mathbf{I})^{-1} (d\mathbf{X}) (\mathbf{X}+z \, \mathbf{I})^{-1} \, dz $$
 * $$\mathbf{X} = \sum_i \lambda_i \mathbf{P}_i$$ is diagonalizable
 * $$ \int_0^\infty (\mathbf{X}+z \, \mathbf{I})^{-1} (d\mathbf{X}) (\mathbf{X}+z \, \mathbf{I})^{-1} \, dz $$
 * $$\mathbf{X} = \sum_i \lambda_i \mathbf{P}_i$$ is diagonalizable
 * $$\mathbf{X} = \sum_i \lambda_i \mathbf{P}_i$$ is diagonalizable
 * $$d \left(f(\mathbf{X})\right) =$$
 * $$\sum_{ij} \mathbf{P}_i (d\mathbf{X}) \mathbf{P}_j \begin{cases}
 * }

To convert to normal derivative form, first convert it to one of the following canonical forms, and then use these identities:

 * {|class="wikitable" style="text-align: center;"

! Canonical differential form !! Equivalent derivative form (numerator layout)
 * + Conversion from differential to derivative form
 * $$dy = a\,dx$$ || $$\frac{dy}{dx} = a$$
 * $$dy = \mathbf{a}^\top d\mathbf{x}$$ || $$\frac{dy}{d\mathbf{x}} = \mathbf{a}^\top$$
 * $$dy = \operatorname{tr}(\mathbf{A}\,d\mathbf{X})$$ || $$\frac{dy}{d\mathbf{X}} = \mathbf{A}$$
 * $$d\mathbf{y} = \mathbf{a}\,dx$$ || $$\frac{d\mathbf{y}}{dx} = \mathbf{a}$$
 * $$d\mathbf{y} = \mathbf{A}\,d\mathbf{x}$$ || $$\frac{d\mathbf{y}}{d\mathbf{x}} = \mathbf{A}$$
 * $$d\mathbf{Y} = \mathbf{A}\,dx$$ || $$\frac{d\mathbf{Y}}{dx} = \mathbf{A}$$
 * }
 * $$d\mathbf{y} = \mathbf{a}\,dx$$ || $$\frac{d\mathbf{y}}{dx} = \mathbf{a}$$
 * $$d\mathbf{y} = \mathbf{A}\,d\mathbf{x}$$ || $$\frac{d\mathbf{y}}{d\mathbf{x}} = \mathbf{A}$$
 * $$d\mathbf{Y} = \mathbf{A}\,dx$$ || $$\frac{d\mathbf{Y}}{dx} = \mathbf{A}$$
 * }
 * $$d\mathbf{Y} = \mathbf{A}\,dx$$ || $$\frac{d\mathbf{Y}}{dx} = \mathbf{A}$$
 * }

Applications
Matrix differential calculus is used in statistics and econometrics, particularly for the statistical analysis of multivariate distributions, especially the multivariate normal distribution and other elliptical distributions.

It is used in regression analysis to compute, for example, the ordinary least squares regression formula for the case of multiple explanatory variables. It is also used in random matrices, statistical moments, local sensitivity and statistical diagnostics.

Software

 * MatrixCalculus.org, a website for evaluating matrix calculus expressions symbolically
 * NCAlgebra, an open-source Mathematica package that has some matrix calculus functionality
 * SymPy supports symbolic matrix derivatives in its matrix expression module, as well as symbolic tensor derivatives in its array expression module.

Information

 * Matrix Reference Manual, Mike Brookes, Imperial College London.
 * Matrix Differentiation (and some other stuff), Randal J. Barnes, Department of Civil Engineering, University of Minnesota.
 * Notes on Matrix Calculus, Paul L. Fackler, North Carolina State University.
 * Matrix Differential Calculus (slide presentation), Zhang Le, University of Edinburgh.
 * Introduction to Vector and Matrix Differentiation (notes on matrix differentiation, in the context of Econometrics), Heino Bohn Nielsen.
 * A note on differentiating matrices (notes on matrix differentiation), Pawel Koval, from Munich Personal RePEc Archive.
 * Vector/Matrix Calculus More notes on matrix differentiation.
 * Matrix Identities (notes on matrix differentiation), Sam Roweis.

title: Matrix calculus


text:
User:Jonesboy1983

Allo--Hypo 21:50, 19 Apr 2005 (UTC)

title: User:Jonesboy1983


text:
User:Caroline Thompson/Bell inequalities/

A Bell inequality is one that must be obeyed under any local hidden variable theory but is violated under quantum mechanics (QM). The term "Bell inequality" can mean any one of a number of inequalities — in practice, in real experiments, the CHSH or  CH74 inequality, not the original one derived by John Bell. It places restrictions on the statistical results of measurements on pairs of particles that are physically separated at the time of measurement but come from a common source and share “hidden variable” values. If the particles have taken part in a quantum interaction, then quantum mechanic predicts that a Bell inequality can be infringed, thus providing an opportunity for an experimental test of QM versus local hidden variable theories, the best known of these being by Alain Aspect and his team in Paris (Aspect, 1981, 1982a,b).

As will be seen from the descriptions below, the various inequalities are not quite equivalent. In practice they involve assumptions that are disputed by somelocal realists, opening the way for loopholes.

Bell's original inequality
The inequality that Bell derived (Bell, 1964) was:

 * $$ 1 + \operatorname{C}(b, c) \geq |\operatorname{C}(a, b) - \operatorname{C}(a, c)|, $$

where C is the "quantum correlation" (the expectation value of the product of the outcomes) of the particle pairs and a, b and c settings of the apparatus (in practice, orientations of the analysing polariser) in a Bell test experiment. Outcomes are coded as +1 for the + channel, -1 for the - channel.

This inequality is not used in practice. For one thing, it is only true for genuinely "two-outcome" systems, not for the "three-outcome" ones (with possible outcomes of zero as well as +1 and -1) encountered in real experiments. For another, it applies only to a very restricted set of hidden variable theories, namely those for which the outcomes on both sides of the experiment are always exactly anticorrelated when the analysers are parallel, in agreement with the quantum mechanical prediction.

Optical experiments, incidentally, can scarcely be said to have confirmed this last prediction, since even with analysers parallel there are a great number of non-detections due to the inefficiency of the detectors.

The CHSH inequality
In 1969 Clauser, Horne, Shimony and Holt (Clauser, 1969) derived the slightly more general "CHSH" inequality, though in the proof they gave initially it would appear to be valid only for two-outcome setups. It was only later, in 1971, that Bell published an improved derivation, valid even when there are non-detections. This derivation will be found on the "CHSH inequality" page.

The CHSH inequality is:

 * $$ -2 \leq \operatorname{C}(a, b) - \operatorname{C}(a, b') + \operatorname{C}(a', b) + \operatorname{C}(a,b) \leq 2 $$

and C is, as before, the quantum correlation when the analyser settings are a and b. Explicitly in terms of the various observed probabilities of pairs this is:

 * $$  \operatorname{C}(a ,b) = p_{++}(a, b) + p_{--}(a, b) - p_{+-}(a, b) - p_{-+}(a, b) $$

This inequality is widely used, though it did not come into favour until after Aspect had inaugurated the idea in 1982 (Aspect, 1982). The reason for reluctance to use it at first was that, strictly speaking, each term C, being formed from sums and differences of four probabilities, ought to be estimated using unbiased estimates of those probabilities. The probabilities ought to be estimated as ratios of the observed frequencies of coincidences to the number N of emitted pairs. This number, however, is not known in real experiments and is, in any case, very large compared to the coincidence counts. If it were to be used, no optical experiment could hope to violate the inequality. So in practice the estimate

 * $$  \operatorname{C}(a b)= \frac{\operatorname{N}_{++}(a,b) + \operatorname{N}_{--}(a,b) - \operatorname{N}_{+-}(a,b) - \operatorname{N}_{-+}(a,b)}{(\operatorname{N}_{++}(a,b) + \operatorname{N}_{--}(a,b) + \operatorname{N}_{+-}(a,b) + \operatorname{N}_{-+}(a,b)} $$

is used. This estimate can only be justified if the sample of detected pairs is a fair sample of those emitted. Local realists believe that there is good reason to expect that the sample is not in fact fair, given the low efficiencies of the detectors and the likely characteristics of the apparatus. The search for more efficient detectors or for other ways of achieving a "loophole-free" experiment continues.

The CH74 inequality
The inequality derived by Clauser and Horne in 1974 depends only on ratios of observed counts, with no presupposition that they represent probability estimates. Knowledge of N, the number of pairs emitted, is not necessary (Clauser, 1974; Clauser, 1978), nor is it important to have high efficiency detectors. Just one assumption is needed in addition to "separability". It is assumed that there is "no enhancement" -- that when a polariser is inserted the probability of detection of a photon never increases. The derivation is given on the page "Clauser and Horne's 1974 Bell test". The inequality is

\frac{\operatorname{N}(a, b) - \operatorname{N}(a, b') + \operatorname{N}(a', b) + \operatorname{N}(a', b') - \operatorname{N}(a', \infty) - \operatorname{N}(\infty, b)}{\operatorname{N}(\infty, \infty)} \leq 0.$$

The Wigner-d'Espagnat inequality
The Wigner-d'Espagnat inequality is almost equivalent to Bell's original inequality, though it is expressed in terms of probabilities, not expectation values of products of outcomes.

Just three different settings are used for the detectors, which are taken as giving definite either + or - outcomes. The inequality is:

 * $$ P_{++} \leq P_{++}(a, c) + P_{++}(c, b) $$

where P is the probability of a coincidence.

Other inequalities
In 1989 Greenberger, Horne, and Zeilinger produced an alternative to the Bell setup, known as the GHZ experiment. It uses three observers and three electrons, and is designed to be able to distinguished a class of hidden variables from quantum mechanics in a single set of observations (Greenberger, 1990, 1993).

In 1993 Hardy proposed a situation where nonlocality can be inferred (in ideal conditions) without using inequalities (Hardy, 1993).

Applications of Bell’s inequalities
See the page Bell test experiments for applications. These experiments are currently believed by most physicists to have provided convincing support for QM and its “nonlocal” prediction of quantum entanglement. has been assumed that the sample of detected pairs is representative of the pairs emitted by the source. That this assumption may not be true comprises the fair sampling loophole. No absolute check on its validity is feasible.


text:
American Comics Group

American Comics Group (ACG) was an American comic book publisher started in 1939 and existing under the ACG name from 1943 to 1967. It published the medium's first ongoing horror-comics title, Adventures into the Unknown. ACG's best-known character was the 1960s satirical-humor hero Herbie Popnecker, who starred for a time in Forbidden Worlds. Herbie would later get his own title and be turned into a "superhero" called the Fat Fury.

Founded by Benjamin W. Sangor, ACG was co-owned by Fred Iger from 1948 to 1967. Iger's father-in-law, Harry Donenfeld, head of National Periodical Publications (later known as DC Comics), was also a co-owner in the early 1960s (though Donenfeld was severely incapacitated and out of the business after an accident in 1962). ACG was distributed by Independent News Company, which also distributed by (and was part of the same company as) DC.

Origins
The company evolved out of a company owned by Sangor. In the mid-1930s, Sangor and Richard E. Hughes began to produce a short-lived prepackaged comics supplement for newspapers. In 1939, the Sangor Shop (as it was informally known) began producing comics for Sangor's son-in-law Ned L. Pines. The Sangor Shop produced the characters and stories of The Black Terror, Pyroman, and Fighting Yank for Pines' Nedor Comics and produced most of the comics for Pines until 1945.

Independent publishing
In 1943, ACG started to publish its own work under such names as B&I Publishing, Michel Publications and Regis Publishing. It acquired the publisher Creston Publications in 1943, making Creston into an ACG imprint. By 1948, it was publishing comics under the name of American Comics Group. Its titles were typical of the times, including horror, crime, mystery, romance, and talking animal comics. In 1948, it began publishing the long-running horror title Adventures into the Unknown. This was the first of a trilogy of ACG horror/supernatural titles that also included Forbidden Worlds (1951–1967) and Unknown Worlds (1960–1967).

In 1949, ACG began publishing two long-running romance titles, Romantic Adventures (later changed to My Romantic Adventures), and Lovelorn (later changed to Confessions of the Lovelorn). Both titles lasted into the 1960s.

The company survived the 1954 Senate subcommittee hearings on the dangers of comic books, even retaining its somewhat diluted horror title Adventures into the Unknown. However, in 1955 ACG canceled four long-running humor titles: the talking-animal series Giggle Comics and Ha Ha Comics, and the teen-humor titles Cookie and The Kilroys.

An October 1, 1952 "Statement of the Ownership, Management, and Circulation" published in ACG's Forbidden Worlds #15 gave its publisher's name as “Preferred Publications, Inc., 8 Lord St., Buffalo, New York” and the owners as Preferred Publications and "B. W. Sangor, 7 West 81st Street, New York, N. Y." The editor was listed as “Richard E. Hughes, 120 West 183rd St., New York, N. Y.” and the business manager as "Frederick H. Iger, 50 Beverly Road, Great Neck, Great Neck, L. I., N. Y." An October 1, 1950, statement published in ACG's Cookie #29 gives identical data, with the exception of the publisher and co-owner being listed as "Michel Publications, Inc. 420 DeSoto Ave., St. Louis 7, Mo.”

Almost all stories after 1957 were written by editor Hughes under a variety of pseudonyms. Besides the satirical superhero the Fat Fury, other ACG superheroes of the period known as the Silver Age of Comic Books included Magicman (starting in Forbidden Worlds #125), Nemesis in Adventures into the Unknown (starting with #154), and John Force, Magic Agent, in his own title in 1962, then later in Unknown Worlds (#35, 36, 48, 50, 52, 56), with a few stories in Forbidden Worlds (#124, 145) and Adventures into the Unknown (#153, 157).

By 1967, the company had ended publication, except for its commercial comics division, Custom Comics, established in 1950, which lasted until the early 1980s doing work for a variety of clients such as the A. C. Gilbert toy company, Montgomery Ward, Tupperware, and the United States Air Force.

Titles

 * Adventures into the Unknown #1–174 (Fall 1948–August 1967)
 * Blazing West #1–20 (Fall 1948–November 1951); continues as The Hooded Horseman v1
 * Commander Battle and the Atomic Sub #1–7 (July 1954–August 1955)
 * Confessions of the Lovelorn #52–114 (63 issues, August 1954–June 1960); continues from Lovelorn (the indicia title through #75)
 * Cookie #1–55 (April 1946–September 1955)
 * Dizzy Dames #1–6 (September 1952–July 1953)
 * Forbidden Worlds #1–145 (July/August 1951–August 1967); paused after #34 to briefly continue as Young Heroes before resuming
 * Funny Films #1–29 (September 1949–May 1954)
 * Gasp! #1–4 (March 1967–August 1967)
 * Giggle Comics #1–99 (October 1943 – January 1955); acquired from Creston Publications; continues as Spencer Spook
 * Ha Ha Comics #1–99 (October 1943 – December 1954/January 1955); continues as TeePee Tim
 * Hi-Jinks #1–7 (July 1947 – July 1948)
 * Herbie #1–23 (April/May 1964 – February 1967)
 * The Hooded Horseman v1 #21–27 (7 issues, January 1952–January 1953); continues from Blazing West
 * The Hooded Horseman v2 #18–22 (5 issues, November 1954–August 1958); continues from Out of the Night
 * The Kilroys #1–54 (June/July 1947–June/July 1955)
 * Lovelorn #1–51 (August/September 1949–July 1954); continues as Confessions of the Lovelorn
 * Magic Agent #1–3 (January/February 1962–May/June 1962)
 * Midnight Mystery (7 issues, January 1961–October 1961)
 * Milt Gross Funnies #1–2 (August 1947–September 1947)
 * Moon Mullins #1–5 (December 1947–August 1948); continued at St. John Publications
 * My Romantic Adventures #49–138 (90 issues, September 1954–March 1964); continues from Romantic Adventures (the indicia title through #71)
 * Operation Peril #1–16 (October 1950–May 1953)
 * Out of the Night (17 issues, February 1952–November 1954); continues as The Hooded Horseman v2
 * Romantic Adventures #1–48 (March/April 1949–August 1954); continues as My Romantic Adventures
 * Search For Love #1–2 (February–April 1950)
 * Skeleton Hand in Secrets of the Supernatural (6 Issues, September/October 1952–July/August 1953)
 * Soldiers of Fortune #1–12 (March/April 1951–January/February 1953) — acquired from Creston Publications
 * Spencer Spook #100–101 (2 issues, March-May 1955); continues from Giggle Comics; see Ace Comics
 * Spy and Counterspy #1–2 (August-October 1949); continues as Spy Hunters
 * Spy Hunters #3–24 (22 issues, December 1949–June 1953); continues from Spy and Counterspy
 * Teepee Tim #101–103 (3 issues, February-June 1955); continues from Ha Ha Comics
 * Unknown Worlds #1–57 (August 1960–August 1967)
 * Wrangler Great Moments in Rodeo (50 issues, 1955–1966)
 * Young Heroes #35–37 (3 issues, February–June 1955); continues from Forbidden Worlds #34
 * One shots: Chuckle (1945), Hi-Jinx (1945), Merry-Go-Round (1944), The Sheriff of Coshise (1957), and The Clutching Hand (1954)

Imprints
Source:
 * B & I Publishing Co., Inc.
 * B. & M. Distributing Co., Inc.
 * Best Syndicated Features, Inc.
 * Creston Publications Corp.
 * Culver Publications
 * Custom Comics, Inc.
 * La Salle Publishing Co.
 * Michel Publications, Inc.
 * Milt Gross, Inc.
 * Modern Store Publications
 * Modern Store Publishing
 * Preferred Publications, Inc.
 * Regis Publications, Inc.
 * Scope Magazines, Inc.
 * Titan Publishing Co. Inc.

title: American Comics Group


text:
Talk:Globals

Marcus Barber's proposed currency
I have blanked the description of Marcus Barber's proposal that water be used as a basis for an international currency. Although he published his paper about this some years ago, no-one has adopted the idea and, since his graduation from Swinburne University of Technology, the paper is no longer published on their web-site. --Theo (Talk) 21:53, 19 Apr 2005 (UTC)

title: Talk:Globals


text:
Wikipedia:Articles for deletion/Pat alton

 This page is an archive of the discussion about the proposed deletion of the article below. This page is no longer live. Further comments should be made on the article's talk page rather than here so that this page is preserved as an historic record. The result of the debate was speedy deleted by SWAdair (Indisputable bad faith addition) --cesarb 16:57, 21 Apr 2005 (UTC)

Pat alton

 * Speedy Delete I was going to do that but the VfD tag was alreadty there. It also looks like someone is having a problem seting up the VfD. Vegaswikian 22:05, 19 Apr 2005 (UTC)
 * It's this newfangled vfd shortcut routine. Driving me haywire. Feel free to speedy. -- 8^D gab 22:06, 2005 Apr 19 (UTC)
 * Delete, not notable, vanity. Megan1967 05:42, 20 Apr 2005 (UTC)
 * Speedily deleted. The deletion log shows it was deleted the same minute that the VfD tag was added (thus unknowingly recreating the article).  I concur with the original deletion, and am deleting it again.  Indisputable bad faith addition.  SWAdair | Talk  06:23, 20 Apr 2005 (UTC)
 * I feel bad for having delayed the deletion this long!!! Sorry about that. -- 8^D gab 16:42, 2005 Apr 20 (UTC)
 * LOL! No problem.  I still remember how confused I was the first time I had to learn a new VfD routine.  It took a while.  SWAdair | Talk  02:43, 21 Apr 2005 (UTC)

 * This page is now preserved as an archive of the debate and, like some other VfD subpages, is no longer 'live'. Subsequent comments on the issue, the deletion, or the decision-making process should be placed on the relevant 'live' pages. Please do not edit this page.


text: User talk:129.89.50.20

title: User talk:129.89.50.20