text:
Symplectic matrix

In mathematics, a symplectic matrix is a $$2n\times 2n$$ matrix $$M$$ with real entries that satisfies the condition

where $$M^\text{T}$$ denotes the transpose of $$M$$ and $$\Omega$$ is a fixed $$2n\times 2n$$ nonsingular, skew-symmetric matrix. This definition can be extended to $$2n\times 2n$$ matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.

Typically $$\Omega$$ is chosen to be the block matrix $$ \Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \\ \end{bmatrix}, $$ where $$I_n$$ is the $$n\times n$$ identity matrix. The matrix $$\Omega$$ has determinant $$+1$$ and its inverse is $$\Omega^{-1} = \Omega^\text{T} = -\Omega$$.

Generators for symplectic matrices
Every symplectic matrix has determinant $$+1$$, and the $$2n\times 2n$$ symplectic matrices with real entries form a subgroup of the general linear group $$\mathrm{GL}(2n;\mathbb{R})$$ under matrix multiplication since being symplectic is a property stable under matrix multiplication. Topologically, this symplectic group is a connected noncompact real Lie group of real dimension $$n(2n+1)$$, and is denoted $$\mathrm{Sp}(2n;\mathbb{R})$$. The symplectic group can be defined as the set of linear transformations that preserve the symplectic form of a real symplectic vector space.

This symplectic group has a distinguished set of generators, which can be used to find all possible symplectic matrices. This includes the following sets $$\begin{align} D(n) =& \left\{ \begin{pmatrix} A & 0 \\ 0 & (A^T)^{-1} \end{pmatrix} : A \in \text{GL}(n;\mathbb{R}) \right\} \\ N(n) =& \left\{ \begin{pmatrix} I_n & B \\ 0 & I_n \end{pmatrix} : B \in \text{Sym}(n;\mathbb{R}) \right\} \end{align}$$ where $$\text{Sym}(n;\mathbb{R})$$ is the set of $$n\times n$$ symmetric matrices. Then, $$\mathrm{Sp}(2n;\mathbb{R})$$ is generated by the set p. 2 $$\{\Omega \} \cup D(n) \cup N(n)$$ of matrices. In other words, any symplectic matrix can be constructed by multiplying matrices in $$D(n)$$ and $$N(n)$$ together, along with some power of $$\Omega$$.

Inverse matrix
Every symplectic matrix is invertible with the inverse matrix given by $$ M^{-1} = \Omega^{-1} M^\text{T} \Omega. $$ Furthermore, the product of two symplectic matrices is, again, a symplectic matrix. This gives the set of all symplectic matrices the structure of a group. There exists a natural manifold structure on this group which makes it into a (real or complex) Lie group called the symplectic group.

Determinantal properties
It follows easily from the definition that the determinant of any symplectic matrix is ±1. Actually, it turns out that the determinant is always +1 for any field. One way to see this is through the use of the Pfaffian and the identity $$ \mbox{Pf}(M^\text{T} \Omega M) = \det(M)\mbox{Pf}(\Omega).$$ Since $$M^\text{T} \Omega M = \Omega$$ and $$\mbox{Pf}(\Omega) \neq 0$$ we have that $$\det(M)=1$$.

When the underlying field is real or complex, one can also show this by factoring the inequality $$\det(M^\text{T} M + I) \ge 1$$.

Block form of symplectic matrices
Suppose Ω is given in the standard form and let $$M$$ be a $$2n\times 2n$$ block matrix given by $$M = \begin{pmatrix}A & B \\ C & D\end{pmatrix}$$ where $$A,B,C,D$$ are $$n\times n$$ matrices. The condition for $$M$$ to be symplectic is equivalent to the two following equivalent conditions "$A^\text{T}C,B^\text{T}D$ symmetric, and $A^\text{T} D - C^\text{T} B = I$""$AB^\text{T},CD^\text{T}$ symmetric, and $AD^\text{T} - BC^\text{T} = I$"The second condition comes from the fact that if $$M$$ is symplectic, then $$M^T$$ is also symplectic. When $$n=1$$ these conditions reduce to the single condition $$\det(M)=1$$. Thus a $$2\times 2$$ matrix is symplectic iff it has unit determinant.

Inverse matrix of block matrix
With $$\Omega$$ in standard form, the inverse of $$M$$ is given by $$ M^{-1} = \Omega^{-1} M^\text{T} \Omega=\begin{pmatrix}D^\text{T} & -B^\text{T} \\-C^\text{T} & A^\text{T}\end{pmatrix}.$$ The group has dimension $$n(2n+1)$$. This can be seen by noting that $$( M^\text{T} \Omega M)^\text{T} = -M^\text{T} \Omega M$$ is anti-symmetric. Since the space of anti-symmetric matrices has dimension $$\binom{2n}{2},$$ the identity $$ M^\text{T} \Omega M = \Omega$$ imposes $$2n \choose 2$$ constraints on the $$(2n)^2$$ coefficients of $$M$$ and leaves $$M$$ with $$n(2n+1)$$ independent coefficients.

Symplectic transformations
In the abstract formulation of linear algebra, matrices are replaced with linear transformations of finite-dimensional vector spaces. The abstract analog of a symplectic matrix is a symplectic transformation of a symplectic vector space. Briefly, a symplectic vector space $$(V,\omega)$$ is a $$2n$$-dimensional vector space $$V$$ equipped with a nondegenerate, skew-symmetric bilinear form $$\omega$$ called the symplectic form.

A symplectic transformation is then a linear transformation $$L:V\to V$$ which preserves $$\omega$$, i.e.
 * $$\omega(Lu, Lv) = \omega(u, v).$$

Fixing a basis for $$V$$, $$\omega$$ can be written as a matrix $$\Omega$$ and $$L$$ as a matrix $$M$$. The condition that $$L$$ be a symplectic transformation is precisely the condition that M be a symplectic matrix:
 * $$M^\text{T} \Omega M = \Omega.$$

Under a change of basis, represented by a matrix A, we have
 * $$\Omega \mapsto A^\text{T} \Omega A$$
 * $$M \mapsto A^{-1} M A.$$

One can always bring $$\Omega$$ to either the standard form given in the introduction or the block diagonal form described below by a suitable choice of A.

The matrix Ω
Symplectic matrices are defined relative to a fixed nonsingular, skew-symmetric matrix $$\Omega$$. As explained in the previous section, $$\Omega$$ can be thought of as the coordinate representation of a nondegenerate skew-symmetric bilinear form. It is a basic result in linear algebra that any two such matrices differ from each other by a change of basis.

The most common alternative to the standard $$\Omega$$ given above is the block diagonal form
 * $$\Omega = \begin{bmatrix}

\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{bmatrix}.$$ This choice differs from the previous one by a permutation of basis vectors.

Sometimes the notation $$J$$ is used instead of $$\Omega$$ for the skew-symmetric matrix. This is a particularly unfortunate choice as it leads to confusion with the notion of a complex structure, which often has the same coordinate expression as $$\Omega$$ but represents a very different structure. A complex structure $$J$$ is the coordinate representation of a linear transformation that squares to $$-I_n$$, whereas $$\Omega$$ is the coordinate representation of a nondegenerate skew-symmetric bilinear form. One could easily choose bases in which $$J$$ is not skew-symmetric or $$\Omega$$ does not square to $$-I_n$$.

Given a hermitian structure on a vector space, $$J$$ and $$\Omega$$ are related via
 * $$\Omega_{ab} = -g_{ac}{J^c}_b$$

where $$g_{ac}$$ is the metric. That $$J$$ and $$\Omega$$ usually have the same coordinate expression (up to an overall sign) is simply a consequence of the fact that the metric g is usually the identity matrix.

Diagonalization and decomposition

 * For any positive definite symmetric real symplectic matrix $S$ there exists $U$ in $$\mathrm{U}(2n,\mathbb{R}) = \mathrm{O}(2n)$$ such that
 * $$S = U^\text{T} D U \quad \text{for} \quad D = \operatorname{diag}(\lambda_1,\ldots,\lambda_n,\lambda_1^{-1},\ldots,\lambda_n^{-1}),$$ where the diagonal elements of $D$ are the eigenvalues of $S$.

 * Any real symplectic matrix $S$ has a polar decomposition of the form:
 * $$S = UR \quad$$ for $$\quad U \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{U}(2n,\mathbb{R})$$ and $$R \in \operatorname{Sp}(2n,\mathbb{R})\cap\operatorname{Sym}_+(2n,\mathbb{R}).$$

such that $O$ and $O'$ are both symplectic and orthogonal and $D$ is positive-definite and diagonal. This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.
 * Any real symplectic matrix can be decomposed as a product of three matrices:

Complex matrices
If instead M is a 2n × 2n matrix with complex entries, the definition is not standard throughout the literature. Many authors adjust the definition above to

where M* denotes the conjugate transpose of M. In this case, the determinant may not be 1, but will have absolute value 1. In the 2×2 case (n=1), M will be the product of a real symplectic matrix and a complex number of absolute value 1.

Other authors retain the definition ($$) for complex matrices and call matrices satisfying ($$) conjugate symplectic.

Applications
Transformations described by symplectic matrices play an important role in quantum optics and in continuous-variable quantum information theory. For instance, symplectic matrices can be used to describe Gaussian (Bogoliubov) transformations of a quantum state of light. In turn, the Bloch-Messiah decomposition ($$) means that such an arbitrary Gaussian transformation can be represented as a set of two passive linear-optical interferometers (corresponding to orthogonal matrices O and O' ) intermitted by a layer of active non-linear squeezing transformations (given in terms of the matrix D). In fact, one can circumvent the need for such in-line active squeezing transformations if two-mode squeezed vacuum states are available as a prior resource only.

title: Symplectic matrix


text:
Soft Cell

Soft Cell are an English synth-pop duo who came to prominence in the early 1980s. The duo consists of vocalist Marc Almond and instrumentalist David Ball. The band are primarily known for their 1981 hit version of "Tainted Love" and their platinum-selling debut album Non-Stop Erotic Cabaret.

In the United Kingdom, Soft Cell had twelve top 40 hits, including "Tainted Love" (number 1), "Torch" (number 2), "Say Hello, Wave Goodbye", "What" (both number 3), and "Bedsitter" (number 4). They also had four top 20 albums between 1981 and 1984. The duo split in 1984 but reformed in 2001 to tour and release a new album. They held a reunion concert in London on 30 September 2018, stating it would be their last live UK performance as a duo but that they may still perform abroad and record together.

Soft Cell's songs have been covered by several artists, including David Gray, Coil and Nine Inch Nails. Their track "Memorabilia" earned recognition for the band as pioneers of the synth-oriented techno genre. The duo have sold in excess of ten million records world-wide.

Mutant Moments and "Memorabilia"
Soft Cell were formed in 1978, after Almond and Ball had met at Leeds Polytechnic in the previous year. Their initial efforts at recording resulted in an EP titled Mutant Moments, released in 1980 and which was funded by a loan of £2,000 from Dave Ball's mother and made with a simple two-track recorder.

Soft Cell's next recording, "The Girl with the Patent Leather Face", appeared as a contribution to the Some Bizzare Album, which included Depeche Mode, the The, and Blancmange. The duo ultimately signed to the Some Bizzare label, backed by Phonogram Records. Their first single, "A Man Could Get Lost" b/w "Memorabilia" on 7" and "Memorabilia" b/w "Persuasion" on 12", was produced by Daniel Miller who founded Mute Records, "Memorabilia" was a success in nightclubs, but did not enter the UK singles chart.

"Tainted Love"
After the chart failure of "Memorabilia", Phonogram Records allowed Soft Cell to record a second and final single in an attempt to score a chart success. The band opted to record a cover version of "Tainted Love", an obscure 1965 northern soul track originally released by Gloria Jones (the girlfriend of Marc Bolan at the time of his death) and written by Ed Cobb of the Four Preps.

Released in 1981, Soft Cell's "Tainted Love" was a No. 1 hit in 17 countries, including the United Kingdom, where it was the second best selling single of 1981 after selling 1.05 million copies (a total which increased to 1.35 million copies by August 2017).

In 1982, "Tainted Love" became a No. 8 single in the United States and went on to set a Guinness World Record at the time for the longest consecutive stay (43 weeks) on the US Billboard Hot 100 chart. The song's popularity developed slowly, needing 19 weeks to enter the US Top 40. The A-side of the 12-inch single of "Tainted Love" actually featured a two-song medley, with "Tainted Love" blending into the Motown classic "Where Did Our Love Go" (originally recorded by the Supremes in 1964).

Non-Stop Erotic Cabaret
The duo's first album, Non-Stop Erotic Cabaret, hit UK No. 5 and further explored the now-trademark Soft Cell themes of squalor and sleaze. "Seedy Films" talks of long nights in porno cinemas, while "Frustration" and "Secret Life" deal with the boredom and hypocrisy associated with suburban life. A companion video titled Non-Stop Exotic Video Show was released in 1982 and featured videos directed by Tim Pope. The video generated some controversy in Britain, mainly due to a scandal involved with the "Sex Dwarf" clip.

Non-Stop Erotic Cabaret garnered two additional hits: "Bedsitter" dealt with the loneliness and lifestyle of a young man having recently left home to live in a bedsit while partying hard. "Bedsitter" reached No. 4 in the UK Singles Chart in November 1981. The song was highly acclaimed in a retrospective review by AllMusic journalist Ned Raggett who wrote that it "ranks as one of the best, most realistic portrayals of urban life recorded." The final single on the album, the ballad "Say Hello, Wave Goodbye", peaked at No. 3 in February 1982.

During 1982, the duo spent most of their time recording and relaxing in New York City, where they met a woman named Cindy Ecstasy who Almond would later confirm was his drug supplier (it was Cindy Ecstasy who introduced them to the new nightclub drug of the same name). Soon after "Say Hello Wave Goodbye" dropped out of the chart, Soft Cell released a brand new song: "Torch", a love song which was to prove the closest the band ever got to having a No. 1 hit with one of their own songs. The song entered straight into the Top 20 and peaked at No. 2.

In June 1982, as was the trend at the time, the duo released a mini album titled Non Stop Ecstatic Dancing, containing remixes of older material along with their new single, "What!", which is a cover of the 1965 song by Melinda Marx, which was later covered in 1968 by Judy Street, whose version became extremely popular on the Northern Soul scene."What!" was a major hit in the UK and reached No. 3 on the UK Singles Chart in August that year.

Decline and dissolution
By 1983, fame and nearly constant drug use were having a bad effect on the duo. Marc Almond also formed the group Marc and the Mambas, featuring collaborations with The The's Matt Johnson and future Almond collaborator Annie Hogan, as an offshoot to experiment out of the glare of the Soft Cell spotlight. Soft Cell's third album release, appropriately titled The Art of Falling Apart, was a Top 5 hit in the UK but the singles were only modest successes. The first single "Where The Heart Is" only reached 21, while the double A-side "Numbers"/"Barriers" peaked at 25. "Numbers" also generated controversy due to references in the song to the drug speed.

In September 1983, the duo released a new single "Soul Inside", which returned them to the UK Top 20, but by early 1984 the duo had amicably decided to end Soft Cell. They played farewell concerts at Hammersmith Palais in January, and released one final album called This Last Night in Sodom (UK No. 12) in March. The album's second and final single "Down in the Subway" (UK No. 24), had already been released in February 1984.

Reunions
Almond and Ball reunited as Soft Cell in 2000 with a series of live dates. They performed at the opening of the Ocean nightclub in London during March 2001. A mini-tour followed later in the year. The track "God Shaped Hole" featured on the Some Bizzare compilation titled I'd Rather Shout at a Returning Echo than Kid Someone's Listening, released during 2001. A new Soft Cell album, Cruelty Without Beauty, was released during late 2002, followed by a European tour and a small US tour during early 2003. The second single from the album, a cover of the Four Seasons' "The Night", reached No. 39 in the UK. Soft Cell had considered recording "The Night" in place of "Tainted Love" during 1981, though keyboardist David Ball stated in a 2003 interview with the BBC, "I think history has kind of shown that we did make the right choice [in 1981]".

In 2005 they released an album compiling some of their early demo recordings called the Bedsit Tapes. Another similar compilation entitled Demo Non Stop was also released around this time.

In November 2008, the band released a remix album entitled Heat. The album included Soft Cell tracks remixed by such acts as Paul Dakeyne, the Grid, Manhattan Clique, Cicada, Richard X, Ladytron, MHC, Atomizer, Mark Moore, Spektrum, The Dark Poets and many more.

In February 2018, Soft Cell announced that they would reunite for one last UK reunion concert. On 21 August 2018, Almond revealed on the Chris Evans Breakfast Show on BBC Radio 2 that Soft Cell were releasing two new tracks ahead of the concert: "Northern Lights" and "Guilty (Cos I Say You Are)". Entitled "Say Hello, Wave Goodbye", the reunion concert took place at the O2 Arena in London on 30 September 2018. It was stated this was their final live UK show as a duo, but that they may still perform abroad and record material together.

In 2019 Ball said a new Soft Cell album was in the works. In May 2019, the To Show You, I've Been There book was released, which came with a vinyl copy of the Magick Mutants EP, a sequel to their debut Mutant Moments EP.

In June 2020, the Mutant Moments EP received its first official release. Originally to be part of the UK June Record Store Day 2020, the 10 inch EP was sold directly through the band's website, as the event was cancelled due to the ongoing COVID-19 pandemic.

In July 2021, the duo announced that their new album would be called Happiness Not Included and would feature tracks such as "Heart Like Chernobyl", "Light Sleepers" and "Nostalgia Machine". The album was released in May 2022, 20 years after their previous studio album release, with Dave Ball revealing Pet Shop Boys were working on a remix for a forthcoming single.

Despite claiming their 2018 reunion show would be the last in the UK, Soft Cell reunited again for 5 performances in 2021. The short tour was to mark 40 years of the Non Stop Erotic Cabaret album, with the band performing the album in full, as well as some hits and new material. The run began at the O2 Academy in Glasgow, Scotland on 10 November and ended at the London Hammersmith Apollo on 16 November. On 16 September 2023 they opened the BBC Radio 2 "Live in the Park" event in Victoria Park Leicester.

Influences
The group cited David Bowie, Marc Bolan and Suicide as primary influences. Almond also named Siouxsie Sioux as a major influence for his early performance style.

Discography

 * Non-Stop Erotic Cabaret (1981)
 * The Art of Falling Apart (1983)
 * This Last Night in Sodom (1984)
 * Cruelty Without Beauty (2002)
 * Happiness Not Included (2022)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Soft Cell


text:
Unitary group

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C), and it has as a subgroup the special unitary group, consisting of those unitary matrices with determinant 1.

In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1, under multiplication. All the unitary groups contain copies of this group.

The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket given by the commutator.

The general unitary group (also called the group of unitary similitudes) consists of all matrices A such that A∗A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.

Unitary groups may also be defined over fields other than the complex numbers. The hyperorthogonal group is an archaic name for the unitary group, especially over finite fields.

Properties
Since the determinant of a unitary matrix is a complex number with norm $1$, the determinant gives a group homomorphism

 * $$\det \colon \operatorname{U}(n) \to \operatorname{U}(1).$$

The kernel of this homomorphism is the set of unitary matrices with determinant $1$. This subgroup is called the special unitary group, denoted $SU(n)$. We then have a short exact sequence of Lie groups:

 * $$1 \to \operatorname{SU}(n) \to \operatorname{U}(n) \to \operatorname{U}(1) \to 1.$$

The above map $U(n)$ to $U(1)$ has a section: we can view $U(1)$ as the subgroup of $U(n)$ that are diagonal with $e^{iθ}$ in the upper left corner and $1$ on the rest of the diagonal. Therefore $U(n)$ is a semidirect product of $U(1)$ with $SU(n)$.

The unitary group $U(n)$ is not abelian for $n > 1$. The center of $U(n)$ is the set of scalar matrices $λI$ with $λ ∈ U(1)$; this follows from Schur's lemma. The center is then isomorphic to $U(1)$. Since the center of $U(n)$ is a $1$-dimensional abelian normal subgroup of $U(n)$, the unitary group is not semisimple, but it is reductive.

Topology
The unitary group U(n) is endowed with the relative topology as a subset of M(n, C), the set of all n × n complex matrices, which is itself homeomorphic to a 2n2-dimensional Euclidean space.

As a topological space, U(n) is both compact and connected. To show that U(n) is connected, recall that any unitary matrix A can be diagonalized by another unitary matrix S. Any diagonal unitary matrix must have complex numbers of absolute value 1 on the main diagonal. We can therefore write

 * $$A = S\,\operatorname{diag}\left(e^{i\theta_1}, \dots, e^{i\theta_n}\right)\,S^{-1}.$$

A path in U(n) from the identity to A is then given by

 * $$t \mapsto S \, \operatorname{diag}\left(e^{it\theta_1}, \dots, e^{it\theta_n}\right)\,S^{-1} .$$

The unitary group is not simply connected; the fundamental group of U(n) is infinite cyclic for all n:

 * $$\pi_1(\operatorname{U}(n)) \cong \mathbf{Z} .$$

To see this, note that the above splitting of U(n) as a semidirect product of SU(n) and U(1) induces a topological product structure on U(n), so that
 * $$\pi_1(\operatorname{U}(n)) \cong \pi_1(\operatorname{SU}(n)) \times \pi_1(\operatorname{U}(1)).$$

Now the first unitary group U(1) is topologically a circle, which is well known to have a fundamental group isomorphic to Z, whereas $$\operatorname{SU}(n)$$ is simply connected.

The determinant map det: U(n) → U(1) induces an isomorphism of fundamental groups, with the splitting U(1) → U(n) inducing the inverse.

The Weyl group of U(n) is the symmetric group Sn, acting on the diagonal torus by permuting the entries:
 * $$\operatorname{diag}\left(e^{i\theta_1}, \dots, e^{i\theta_n}\right) \mapsto \operatorname{diag}\left(e^{i\theta_{\sigma(1)}}, \dots, e^{i\theta_{\sigma(n)}}\right)$$

2-out-of-3 property
The unitary group is the 3-fold intersection of the orthogonal, complex, and symplectic groups:

 * $$\operatorname{U}(n) = \operatorname{O}(2n) \cap \operatorname{GL}(n, \mathbf{C}) \cap \operatorname{Sp}(2n, \mathbf{R}) .$$

Thus a unitary structure can be seen as an orthogonal structure, a complex structure, and a symplectic structure, which are required to be compatible (meaning that one uses the same J in the complex structure and the symplectic form, and that this J is orthogonal; writing all the groups as matrix groups fixes a J (which is orthogonal) and ensures compatibility).

In fact, it is the intersection of any two of these three; thus a compatible orthogonal and complex structure induce a symplectic structure, and so forth.

At the level of equations, this can be seen as follows:
 * $$\begin{array}{r|r}

\text{Symplectic} & A^\mathsf{T}JA = J \\ \hline \text{Complex} & A^{-1}JA = J \\ \hline \text{Orthogonal} & A^\mathsf{T} = A^{-1} \end{array}$$

Any two of these equations implies the third.

At the level of forms, this can be seen by decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). On an almost Kähler manifold, one can write this decomposition as $h = g + iω$, where $h$ is the Hermitian form, $g$ is the Riemannian metric, $i$ is the almost complex structure, and $ω$ is the almost symplectic structure.

From the point of view of Lie groups, this can partly be explained as follows: O(2n) is the maximal compact subgroup of GL(2n, R), and U(n) is the maximal compact subgroup of both GL(n, C) and Sp(2n). Thus the intersection O(2n) ∩ GL(n, C) or O(2n) ∩ Sp(2n) is the maximal compact subgroup of both of these, so U(n). From this perspective, what is unexpected is the intersection GL(n, C) ∩ Sp(2n) = U(n).

Special unitary and projective unitary groups

Just as the orthogonal group O(n) has the special orthogonal group SO(n) as subgroup and the projective orthogonal group PO(n) as quotient, and the projective special orthogonal group PSO(n) as subquotient, the unitary group U(n) has associated to it the special unitary group SU(n), the projective unitary group PU(n), and the projective special unitary group PSU(n). These are related as by the commutative diagram at right; notably, both projective groups are equal: PSU(n) = PU(n).

The above is for the classical unitary group (over the complex numbers) – for unitary groups over finite fields, one similarly obtains special unitary and projective unitary groups, but in general $$\operatorname{PSU}\left(n, q^2\right) \neq \operatorname{PU}\left(n, q^2\right)$$.

G-structure: almost Hermitian
In the language of G-structures, a manifold with a U(n)-structure is an almost Hermitian manifold.

Generalizations
From the point of view of Lie theory, the classical unitary group is a real form of the Steinberg group $${}^2\!A_n$$, which is an algebraic group that arises from the combination of the diagram automorphism of the general linear group (reversing the Dynkin diagram An, which corresponds to transpose inverse) and the field automorphism of the extension C/R (namely complex conjugation). Both these automorphisms are automorphisms of the algebraic group, have order 2, and commute, and the unitary group is the fixed points of the product automorphism, as an algebraic group. The classical unitary group is a real form of this group, corresponding to the standard Hermitian form Ψ, which is positive definite.

This can be generalized in a number of ways:
 * generalizing to other Hermitian forms yields indefinite unitary groups U(p, q);
 * the field extension can be replaced by any degree 2 separable algebra, most notably a degree 2 extension of a finite field;
 * generalizing to other diagrams yields other groups of Lie type, namely the other Steinberg groups $${}^2\!D_n, {}^2\!E_6, {}^3\!D_4,$$ (in addition to $${}^2\!A_n$$) and Suzuki-Ree groups
 * $${}^2\!B_2\left(2^{2n+1}\right), {}^2\!F_4\left(2^{2n+1}\right), {}^2\!G_2\left(3^{2n+1}\right);$$
 * considering a generalized unitary group as an algebraic group, one can take its points over various algebras.

Indefinite forms
Analogous to the indefinite orthogonal groups, one can define an indefinite unitary group, by considering the transforms that preserve a given Hermitian form, not necessarily positive definite (but generally taken to be non-degenerate). Here one is working with a vector space over the complex numbers.

Given a Hermitian form Ψ on a complex vector space V, the unitary group U(Ψ) is the group of transforms that preserve the form: the transform M such that Ψ(Mv, Mw) = Ψ(v, w) for all v, w ∈ V. In terms of matrices, representing the form by a matrix denoted Φ, this says that M∗ΦM = Φ.

Just as for symmetric forms over the reals, Hermitian forms are determined by signature, and are all unitarily congruent to a diagonal form with p entries of 1 on the diagonal and q entries of −1. The non-degenerate assumption is equivalent to p + q = n. In a standard basis, this is represented as a quadratic form as:
 * $$\lVert z \rVert_\Psi^2 = \lVert z_1 \rVert^2 + \dots + \lVert z_p \rVert^2 - \lVert z_{p+1} \rVert^2 - \dots - \lVert z_n \rVert^2$$

and as a symmetric form as:
 * $$\Psi(w, z) = \bar w_1 z_1 + \cdots + \bar w_p z_p - \bar w_{p+1}z_{p+1} - \cdots - \bar w_n z_n.$$

The resulting group is denoted U(p,q).

Finite fields
Over the finite field with q = pr elements, Fq, there is a unique quadratic extension field, Fq2, with order 2 automorphism $$\alpha\colon x \mapsto x^q$$ (the rth power of the Frobenius automorphism). This allows one to define a Hermitian form on an Fq2 vector space V, as an Fq-bilinear map $$\Psi\colon V \times V \to K$$ such that $$\Psi(w, v) = \alpha \left(\Psi(v, w)\right)$$ and $$\Psi(w, cv) = c\Psi(w, v)$$ for c ∈ Fq2. Further, all non-degenerate Hermitian forms on a vector space over a finite field are unitarily congruent to the standard one, represented by the identity matrix; that is, any Hermitian form is unitarily equivalent to
 * $$\Psi(w, v) = w^\alpha \cdot v = \sum_{i=1}^n w_i^q v_i$$

where $$w_i,v_i$$ represent the coordinates of w, v ∈ V in some particular Fq2-basis of the n-dimensional space V.

Thus one can define a (unique) unitary group of dimension n for the extension Fq2/Fq, denoted either as U(n, q) or U(n, q2) depending on the author. The subgroup of the unitary group consisting of matrices of determinant 1 is called the special unitary group and denoted SU(n, q) or SU(n, q2). For convenience, this article will use the U(n, q2) convention. The center of U(n, q2) has order q + 1 and consists of the scalar matrices that are unitary, that is those matrices cIV with $$c^{q+1} = 1$$. The center of the special unitary group has order gcd(n, q + 1) and consists of those unitary scalars which also have order dividing n. The quotient of the unitary group by its center is called the projective unitary group, PU(n, q2), and the quotient of the special unitary group by its center is the projective special unitary group PSU(n, q2). In most cases (n > 1 and (n, q2) ∉ {(2, 22), (2, 32), (3, 22)}), SU(n, q2) is a perfect group and PSU(n, q2) is a finite simple group,.

Degree-2 separable algebras
More generally, given a field k and a degree-2 separable k-algebra K (which may be a field extension but need not be), one can define unitary groups with respect to this extension.

First, there is a unique k-automorphism of K $$a \mapsto \bar a$$ which is an involution and fixes exactly k ($$a = \bar{a}$$ if and only if a ∈ k). This generalizes complex conjugation and the conjugation of degree 2 finite field extensions, and allows one to define Hermitian forms and unitary groups as above.

Algebraic groups
The equations defining a unitary group are polynomial equations over k (but not over K): for the standard form $Φ = I$, the equations are given in matrices as $A^{∗}A = I$, where $$A^* = \bar{A}^\mathsf{T}$$ is the conjugate transpose. Given a different form, they are $A^{∗}ΦA = Φ$. The unitary group is thus an algebraic group, whose points over a k-algebra R are given by:

 * $$\operatorname{U}(n, K/k, \Phi)(R) := \left\{ A \in \operatorname{GL}(n, K \otimes_k R) : A^*\Phi A = \Phi\right\}.$$

For the field extension C/R and the standard (positive definite) Hermitian form, these yield an algebraic group with real and complex points given by:

 * $$\begin{align}

\operatorname{U}(n, \mathbf{C}/\mathbf{R})(\mathbf{R}) &= \operatorname{U}(n) \\ \operatorname{U}(n, \mathbf{C}/\mathbf{R})(\mathbf{C}) &= \operatorname{GL}(n, \mathbf{C}). \end{align}$$

In fact, the unitary group is a linear algebraic group.

Unitary group of a quadratic module
The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined, which incorporates as special cases many different classical algebraic groups. The definition goes back to Anthony Bak's thesis.

To define it, one has to define quadratic modules first:

Let R be a ring with anti-automorphism J, $$\varepsilon \in R^\times$$ such that $$r^{J^2} = \varepsilon r \varepsilon^{-1}$$ for all r in R and $$\varepsilon^J = \varepsilon^{-1}$$. Define

 * $$\begin{align}

\Lambda_\text{min} &:= \left\{r \in R \ : \ r - r^J\varepsilon\right\}, \\ \Lambda_\text{max} &:= \left\{r \in R \ : \ r^J\varepsilon = -r\right\}. \end{align}$$

Let $Λ ⊆ R$ be an additive subgroup of R, then Λ is called form parameter if $$\Lambda_\text{min} \subseteq \Lambda \subseteq \Lambda_\text{max}$$ and $$r^J \Lambda r \subseteq \Lambda$$. A pair $(R, Λ)$ such that R is a ring and Λ a form parameter is called form ring.

Let M be an R-module and f a J-sesquilinear form on M (i.e., $$f(xr, ys) = r^J f(x, y)s$$ for any $$x, y \in M$$ and $$r, s \in R$$). Define $$h(x, y) := f(x, y) + f(y, x)^J \varepsilon \in R$$ and $$q(x) := f(x, x) \in R/\Lambda$$, then f is said to define the Λ-quadratic form $(h, q)$ on M. A quadratic module over $(R, Λ)$ is a triple $(M, h, q)$ such that M is an R-module and $(h, q)$ is a Λ-quadratic form.

To any quadratic module $(M, h, q)$ defined by a J-sesquilinear form f on M over a form ring $(R, Λ)$ one can associate the unitary group

 * $$U(M) := \{\sigma \in GL(M) \ : \ \forall x, y \in M, h(\sigma x, \sigma y) = h(x, y) \text{ and } q(\sigma x) = q(x) \}.$$

The special case where $Λ = Λ_{max}$, with J any non-trivial involution (i.e., $$J \neq id_R, J^2 = id_R$$ and $ε = −1$ gives back the "classical" unitary group (as an algebraic group).

Polynomial invariants
The unitary groups are the automorphisms of two polynomials in real non-commutative variables:

 * $$\begin{align}

C_1 &= \left(u^2 + v^2\right) + \left(w^2 + x^2\right) + \left(y^2 + z^2\right) + \ldots \\ C_2 &= \left(uv - vu\right) + \left(wx - xw\right) + \left(yz - zy\right) + \ldots \end{align}$$

These are easily seen to be the real and imaginary parts of the complex form $$Z \overline{Z}$$. The two invariants separately are invariants of O(2n) and Sp(2n). Combined they make the invariants of U(n) which is a subgroup of both these groups. The variables must be non-commutative in these invariants otherwise the second polynomial is identically zero.

Classifying space
The classifying space for U(n) is described in the article Classifying space for U(n).

title: Unitary group


text:
Ferdinand Philippe, Duke of Orléans

Ferdinand Philippe, Duke of Orléans (Ferdinand Philippe Louis Charles Henri Joseph; 3 September 1810 – 13 July 1842) was the eldest son of King Louis Philippe I of France and Maria Amalia of Naples and Sicily. He was born in exile in his mother's native Sicily while his parents were the Duke and Duchess of Orléans. Ferdinand Philippe was heir to the House of Orléans from birth. Following his father's succession as King of the French in 1830, he became the Prince Royal of France and Duke of Orléans. He died in 1842, never to succeed his father or see the collapse of the July Monarchy and subsequent exile of his family to the United Kingdom.

Early life

Born in Palermo in September 1810, during his parents' exile, he was given the title Duke of Chartres (and was called Chartres within the family circle). He was baptised Ferdinand Philippe Louis Charles Henri Joseph and known as Ferdinand Philippe in honour of his grandfathers, Ferdinand I of the Two Sicilies and Philippe Égalité. Despite having been born in exile, he held the rank of prince of the blood and was styled Serene Highness. As the eldest son, he was the heir to the title of Duke of Orléans, head of the House of Orléans (a cadet branch of the Bourbons of France descended from the only brother of Louis XIV).

The young prince first visited France in 1814 during the First Restoration, settling there more permanently in 1817. In 1819 his father put him in the care of a tutor, M. de Boismilon, at the Collège Henri-IV. Louis Philippe wished his son to receive a liberal education on a foundation of complete equality with his fellow students. Ferdinand Philippe was highly successful in his studies and took courses at the École polytechnique. After a trip to Great Britain (visiting both England and Scotland) in 1819, he went to Lunéville to join the 1er régiment de hussards, of which he was made colonel by Charles X in 1824. In September 1824, King Charles X granted him the style "Royal Highness", a style maintained by Ferdinand Philippe at his father's accession to the throne six years later.

July Revolution
In 1830, during the July Revolution, the young Duke of Chartres was on garrison duty at Joigny. He made his regiment wear the tricolor cockade and quickly led them to aid the uprising in Paris. He was temporarily stopped at Montrouge, and entered Paris on 3 August at the head of his regiment. When his father was offered the French throne by the Chamber of Deputies, Prince Ferdinand Philippe received the title of Duke of Orléans, Prince of Orléans, and also became Prince Royal, the heir apparent to the throne. Upon entering the Conseil (at his father's bidding), Ferdinand Philippe, who had something of a temper, criticised the time lost by ministers' prevarications and was frequently embroiled in skirmishes with the doctrinaires, to whom he wished to impart the sentiments of revolutionary youth. When Casimir Périer was nominated president of the Conseil in March 1831, he accepted the post only on condition that Ferdinand Philippe be excluded from the Conseil.

In November 1831, the young Duke of Orléans and Marshal Jean-de-Dieu Soult were sent to repress the Canut revolts. He acquitted himself of this difficult task without violence and managed to rapidly appease opponents of the July Monarchy, even gaining a certain popularity. During the cholera outbreak in 1832, he did not hesitate to take real risks in visiting the most sickly patients at the Hôtel-Dieu de Paris, accompanied by Casimir Périer (who caught the disease and died). In the eyes of the people and the press he was seen as a generous prince, sincerely preoccupied with the plight of the poor, and he became a sort of icon for the dynastic opposition of politician Odilon Barrot, who saw in him the only prince capable of reconciling modern France's democratic aspirations with the heritage of its monarchical past. On 2 March 1832 he was granted an annual income of 1 million francs under his father's new Civil List.

Military career

In 1831, under Marshal Étienne Maurice Gérard, Ferdinand Philippe and his young brother Prince Louis, Duke of Nemours, set out on their first campaign. When the princes entered Belgium in 1831, they eagerly visited the plain of Jemmapes, where their father had fought in 1792. The following year, Ferdinand Philippe returned to Belgium in command of the vanguard brigade of the Armée du Nord. On 20 November 1832 he was before the citadel of Antwerp, and commanded the trenches on the night of 29/30 November. During the attack on the Saint-Laurent Lunette, he launched himself onto the parapet amidst a hail of projectiles to lead the action and arouse his soldiers' courage.

In 1835, when Marshal Bertrand Clauzel was sent to Algeria as Governor General, the young Prince Royal asked his father for permission to accompany him, so he could fight the Emir Abd El-Kader. He participated with Clauzel's army in the Battle of Habrah, where he was wounded, and in the capture of Mascara in December 1835. He then participated in the taking of Tlemcen in January 1836. When he returned to Paris, it was with an aura of military glory, and he returned to Algeria in autumn 1839 to take possession of the country's interior (from Constantine to Algiers) for France alongside Marshal Sylvain Charles Valée. He left Constantine on 16 October, three days after the second anniversary of the town's capture, and reached Algiers on 2 November via Sétif and the Iron Gates pass. Abd-el-Kader saw this as a violation of the Treaty of Tafna and unleashed jihad upon the French. This led to an escalation in tension and, ultimately, Algeria's wholesale occupation by France. Ferdinand Philippe set out for Algeria a third time in March 1840, taking with him his younger brother the Duke of Aumale, tutoring him in his first military experience. Present at the battles of Affroun, Oued'Ger and Bois des Oliviers, he was put in charge of directing the attackers in the capture of Teniah de Mouzaïa. After this campaign he was recalled to France for good.

This brilliant military career increased his popularity and prestige. He also devoted himself to the improvement of the troops' living conditions and morale. At Saint-Omer he organised the chasseurs de Vincennes, who became the chasseurs d'Orléans in 1836, and re-formed the chasseurs à pied de Vincennes. He laid the foundations for a Histoire des Régiments, commissioned by order of the Minister of War, and began writing the service histories of the two regiments he had himself commanded.

Marriage negotiations
Ferdinand Philippe's marriage had long been one of the July Monarchy's major political affairs. Had it not been for the 1830 Revolution, he would have married the sister of Henri, Count of Chambord, Louise d'Artois (1819–1864). Her family called off the marriage plans when Ferdinand Philippe's branch of the family 'usurped' the throne. From 1835, after an assassination attempt by Giuseppe Fieschi and his co-conspirators, Ferdinand Philippe's father had been obsessed with the marital prospects of his son, then 25. Blanche-Joséphine Le Bascle d'Argenteuil noted, in her Souvenirs, that if the Prince Royal died young after having fathered a male heir, the July Monarchy would be faced with the prospect of a regency, in all its political uncertainty – thus, for her, the wisest course consisted of marrying off the King's third son, then the fourth, then the fifth, guaranteeing Louis Philippe descendants, all the while leaving several men around the throne who could take over from him if he died suddenly.

At this time the July Monarchy was searching for new allies in Continental Europe so they would not have to depend solely on the United Kingdom. Talleyrand, fresh from renouncing his embassy to London and close to the British Foreign Secretary, Lord Palmerston, pointed in this direction. The King at first envisaged a rapprochement with Russia via Württemberg. King Wilhelm I, widower of Grand Duchess Catherine Pavlovna of Russia, had two daughters of marriageable age, Princesses Marie (born 1816) and Sophie (born 1818). Wilhelm I's sister Catharina had already made an inauspicious French marriage alliance to Jérôme Bonaparte and so Wilhelm declined the proposition as humiliating. He later was to accept Marie's even more humiliating marriage to Count Alfred von Neipperg in 1840. Queen Louise wrote to her parents on Marie's marriage that "We see singular things. It was not at all probable that this daughter, who the king of Württemberg did not wish to give to Chartres for fear of his ending his days [in exile] in America, should end up marrying a miserable little Austrian officer without illustriousness and of very ordinary birth."

Louis Philippe next envisaged an alliance with Austria via the marriage of his son to Archduchess Maria Theresa (born 1816), daughter of Archduke Karl, Duke of Teschen (German: Herzog von Teschen). Queen Marie Amélie was highly favourable to such a match as she was herself a daughter of an Austrian archduchess (Maria Carolina of Austria), and Archduke Karl was not opposed to it. However, Karl faced determined opponents on two sides – Prince Metternich, who did not want to repeat his error in marrying Marie Louise to Napoleon, and Archduchess Sophie, a Bavarian princess and sister-in-law of the new Kaiser Ferdinand I, who dominated the Vienna court with her strong personality, and was awaiting her son Franz-Josef's ascent to the imperial throne. France's ambassador to Vienna, the Count of Sainte-Aulaire, who had been put in charge of preparing the ground for an Austrian match, felt the possibility was difficult if not completely impossible. The new President of the Conseil, Adolphe Thiers, dreamed of concluding such a match and becoming a new Duke of Choiseul as the maker of a spectacular reversal in the alliances of Europe.

Ferdinand Philippe and his younger brother, the Prince Louis, Duke of Nemours, set out on a European tour on 2 May 1836. Ferdinand Philippe and Queen Marie-Amélie got off to a bad start when the young French Prince Royal refused to shave off the proud beard that had set a fashion among French youth. She wrote back to Queen Louise complaining that "there was a lack of tact there and of sentiments of convenience that afflicted me. [...] I believe that Leopold I can say to him that a goatee beard on the face of a prince royal is contrary to all German manners. Here, [such a beard] is neither handsome nor fortunate, there it can be fatal." The two French princes were a great success in Berlin and Vienna, staying at the latter from 29 May to 11 June. However, the Marquis de Sémonville commented that "everyone has shaken their hand, but no one was close to them". Even if the young Prince Royal decidedly liked Archduke Karl and his daughter, Prince Metternich and the Archduchess Sophie put up a major barrage of problems, and news of Thiers (impatient to conclude the match) being on his way was enough to convince Louis Philippe to make a marriage proposal. He was refused, though to play to French susceptibilities the official version was that the refusal was because of the "feelings" of Archduchess Marie-Thérèse. Queen Louise wrote to her mother on 14 June 1836: "I am upset to see that you have thrown your all behind the cause of Austria [...] I have always thought that Chartres was of too high birth to marry she who seems to him the most minor princess in Germany; and I avow that I would better like to see him marry a princess from Lippe or Waldeck who was good and pretty and of robust health, rather than an archduchess of Austria who would bring us all sorts of evils in her dowry. [...] Napoleon, in this situation, was able to make sacrifices to ally himself with Austria; and we all saw what profit he got from it. But we are not upstarts, and have no need of ennobling ourselves by uniting with the house of Lorraine".

The two young princes returned to France via Italy. At Trento they were received by Marie Louise, Duchess of Parma, the former Empress Consort of the French, who could not refrain from tears at the similarity between the Prince Royal and her son, the late Duke of Reichstadt. At Milan they stayed with Archduke Rainer Joseph of Austria, Viceroy of Lombardy-Venetia, where they heard the news of Alibaud's assassination attempt on King Louis Philippe on 25 June. After the Austrians' refusal of the match, only two potential Catholic princesses remained (Louis-Philippe confided to one of his familiars "I would prefer her to be a Catholic. You believe it is nothing, the Carlists believe it is everything; and I myself believe that it is neither here nor there" ), and these were both very young for marriage (born in 1821): Princess Januaria of Brazil, daughter of Emperor Pedro I of Brazil, and the Infanta Isabella of Spain, daughter of the Infante Francisco de Paula, younger brother of King Ferdinand VII. The former was excluded by her remoteness, and the latter due to her family's unfortunate history (her mother Princess Luisa Carlotta of the Two Sicilies, niece of Queen Marie-Amélie, was monstrously obese) and her physical appearance (she was red-haired and thin; Queen Louise wrote to Queen Marie-Amélie on 21 November 1836 that "I send you her portrait, that Leopold found hideous. Her hair especially is frightening in terms of the children she will have. If all her family are ginger, this will afflict them [too]".

Marriage

Some possibilities were also seen among the Protestant German princesses. Via his great-niece the Duchess of Dino, Talleyrand suggested Princess Louise of Hesse-Kassel (born 1817 to a cousin of the Elector of Hesse and his wife, a Danish princess), whilst Queen Louise suggested Princess Marie of Saxe-Altenburg (born 1818 to the Duke of Saxe-Altenburg and Princess Amelia of Wurtemberg, and who finally ended up marrying King George V of Hanover in 1843), and Princess Victoria of Saxe-Coburg and Gotha (daughter of an elder brother of King Leopold I of the Belgians; she was actually raised a Catholic and married the Prince Royal's younger brother, the Duke of Nemours, in 1840).

However, the negotiators' choice finally came to rest on Duchess Helene Luise Elisabeth of Mecklenburg-Schwerin (known as Hélène, 1814–1858), daughter of the late Prince Frederick Louis, Hereditary Grand Duke of Mecklenburg-Schwerin, and his wife Princess Caroline Louise of Saxe-Weimar-Eisenach. For Ferdinand Philippe, it was a convenient alliance but one without much attraction; Metternich quipped that she was "petite but of a good house". She was the niece of King Friedrich Wilhelm III of Prussia, whose wife was born Louise of Mecklenburg-Strelitz (this did not avoid difficulties for the marriage in Berlin, which the French ambassador there, the Count Bresson, succeeded in resolving). Nicholas I of Russia, for his part, affected disdain of the marriage, proclaiming that such a minor marriage was not worth the trouble to prevent.

The Duke of Broglie was sent to Germany as ambassador extraordinary with the aim of presenting the official marriage request and bringing the princess back to France. An anonymous but virulent libel against the House of Orléans was published by a prince of the House of Mecklenburg. That House avoided the marriage, so that Duchess Hélène was only accompanied to France by her father's widow, Augusta of Hesse-Homburg. The marriage was celebrated on 30 May 1837 at the Château de Fontainebleau, since Hyacinthe-Louis de Quélen, the Archbishop of Paris, had used the pretext of religious differences to forbid it from taking place in Notre Dame de Paris. The civil ceremony occurred in the galerie Henri II on 30 May 1837, presided over by the Baron Pasquier, whom the King rewarded on 27 May by making him Lord Chancellor of France. The Catholic ceremony was presided over by Romain-Frédéric Gallard, Bishop of Meaux, in the chapel of Henri IV, whilst the Lutheran one was celebrated by Pastor Cuvier in the Salon Louis Philippe. As his witnesses, Ferdinand Philippe had the four vice-presidents of the Chamber of Peers: Baron Séguier, Count Portalis, the Duke of Broglie, and the Count de Bastard; the president and four vice presidents of the Chamber of Deputies: Dupin, Jean-Louis Calmon, Delessert, Jacqueminot, Cunin-Gridaine; three marshals: the Duke of Dalmatie, the Count de Lobau and Gérard, as well as the Prince de Talleyrand, the Duke of Choiseul, and Count Bresson, France's minister to Berlin.

The ceremony was well attended, but there was a notable lack of foreign ambassadors, except for Baron von Werther (Prussia), Count Le Hon (Belgium), and the chargé d'affaires of Mecklembourg. The reception was brilliant; the Duchesse de Maillé observed: Princess Hélène was not a king's daughter, and so the model for [the ceremonies] was the reception for Madame the Duchess of Burgundy, and all that happened in the house of Sa Majesté citoyenne was as if Louis XIV was present amidst the most major lords of France. Some believed that Louis Philippe made a political mistake. I think not. To the contrary, he greatly pleased his supporters. The pomp did not displease those whose names figured in it, in place of the great lords who so envied them. Louis Philippe was the man of the middle class, elected by them, and they know that full well, but they were flattered by the shine in which he surrounded himself. If he did not seek to regild this kingdom that [the middle class] has given him, its self-respect would be wounded. His supporters thought themselves great lords when they saw a great king.

The marriage was very happy and produced two children:

 * 1) Prince Philippe, Count of Paris (1838–1894), Prince Royal, married Princess Marie Isabelle of Orléans, Infanta of Spain (1848–1919), and had issue.
 * 2) Prince Robert, Duke of Chartres (1840–1910), married Princess Françoise of Orléans (1844–1925) and had issue.

Patron of the arts

Ferdinand Philippe loved literature, music, and the fine arts, and had a pronounced taste for collecting, "making his choice slowly, like a true lover [of the arts]". Each year he spent 100,000 to 150,000 francs from his Civil List allowance on art purchases or cultural patronage. In his vast apartments in the Palais des Tuileries he gathered medieval and Renaissance objects, ceramics by Bernard Palissy, Hispano-Moorish majolica and ceramics, Chinese and Japanese porcelain, and furniture by Caffieri, Oeben, Riesener, and Jacob. He was passionate about modern painters, buying several canvasses from Ary Scheffer and Newton Fielding, both of whom had taught Ferdinand Philippe landscape painting from 1822 to 1830. He possessed works by Ferdinand-Victor-Eugène Delacroix (The Prisoner of Chillon, The Assassination of the Bishop of Liège, Hamlet and Horatio in the Graveyard), Alexandre-Gabriel Decamps (The Defeat of the Cimbri), Eugène Lami, Ernest Meissonnier, and Paul Delaroche. He loved landscapes by painters of the Barbizon school, notably Camille Corot, Paul Huet, and Théodore Rousseau. He commissioned Jean-Auguste-Dominique Ingres to paint Antiochus and Stratonice (1833), bought his Œdipus and the Sphinx in 1839, and commissioned his portrait from him in 1840.

Himself a talented draughtsman, Ferdinand Philippe made amateur engravings – twelve etchings and lithographs by him are known, including a satire showing the sleeping Gulliver with Lilliputians all round him on foot and on horseback and a sign referring to the alarmist proclamation of 11 July 1792 by the Legislative Assembly that declared the fatherland to be in danger.

Death
In 1842, the Duke was scheduled to leave for Saint-Omer to review part of the army of which he had been made the commander that was engaged at the Marne. He planned to travel from the Tuileries Palace to Neuilly-sur-Seine on 13 July 1842 to say goodbye to his family, and, for the sake of expediency, opted for an open carriage. When the horses of his carriage ran out of control at Sablonville in the Hauts-de-Seine département, he lost his balance and fractured his skull, and, despite the best attentions of his doctors, the 31-year-old Duke died some hours later, surrounded by family members who had rushed to the scene. Alfred de Musset evoked the accident in his poem Le Treize Juillet (in the collection Poésies nouvelles).

Ferdinand Philippe's funeral service was held in Notre Dame which was covered not in black fabric (of which there wasn't enough) but, at the suggestion of the architect Visconti, black paper. He was interred in an elaborate tomb in the Chapelle Royale, in Dreux, Eure-et-Loir.

Deprived of the popular support his eldest son had had, Louis Philippe and his regime fell six years later. He, his family, and Ferdinand Philippe's widow Princess Hélène went into exile in Great Britain. There, Hélène died nearly 16 years after her husband, on 18 May 1858 in Richmond, Surrey. Because Hélène was a Protestant, she could not be buried in the Catholic Chapelle Royale at Dreux. Instead, a room with a separate entrance was built attached to the chapel and a window was opened between her tomb and her husband's. The sculpture of the Protestant princess rests atop her tomb, depicting her reaching through the opening to the tomb of her beloved Catholic prince and husband Ferdinand Philippe.

Honours

 * Kingdom of France: Knight of the Holy Spirit, 30 May 1825
 * 🇧🇪 Belgium: Grand Cordon of the Order of Leopold, 10 March 1833
 * Two Sicilies:
 * Knight of St. Januarius
 * Grand Cross of St. Ferdinand and Merit
 * 🇪🇸 Spain: Knight of the Golden Fleece, 1 July 1837
 * Kingdom of Portugal: Grand Cross of the Tower and Sword

In fiction

 * Ferdinand-Philippe was used by Hanns Heinz Ewers as a character in his novella "Die Herzen der Könige" (The Hearts of the Kings).

title: Ferdinand Philippe, Duke of Orléans


text:
Jeremy Taylor

Jeremy Taylor (1613–1667) was a cleric in the Church of England who achieved fame as an author during the Protectorate of Oliver Cromwell. He is sometimes known as the "Shakespeare of Divines" for his poetic style of expression, and he is frequently cited as one of the greatest prose writers in the English language.

Taylor was under the patronage of William Laud, Archbishop of Canterbury. He went on to become chaplain in ordinary to King Charles I as a result of Laud's sponsorship. This made him politically suspect when Laud was tried by Parliament and executed in January 1644/5 during the English Civil War. After the parliamentary victory over the King, he was briefly imprisoned several times.

Eventually, he was allowed to live quietly in Wales, where he became the private chaplain of the Earl of Carbery. After the Restoration, he was made Bishop of Down and Connor in Ireland. He also became Vice-Chancellor of the University of Dublin.

He is remembered in the liturgical calendars of the Church of England and other Anglican churches.

Early life
Taylor was born in Cambridge, the son of a barber, Nathaniel. He was baptised, as a child, on 15 August 1613 at Holy Trinity Church, Cambridge. His father was educated and taught him grammar and mathematics. He was then educated at the Perse School, Cambridge, before going to Gonville and Caius College, Cambridge, where he gained a Bachelor of Arts degree in 1630/1631 and a Master of Arts degree in 1634.

The best evidence of his diligence as a student is the enormous learning of which he showed so easy a command in later years. In 1633, although still below the canonical age, he took holy orders, and accepted the invitation of Thomas Risden, a former fellow student, to supply his place for a short time as lecturer at St Paul's Cathedral.

Career under Laud
Archbishop William Laud sent for Taylor to preach in his presence at Lambeth, and took the young man under his wing. Taylor did not vacate his fellowship at Cambridge before 1636, but he spent, apparently, much of his time in London, for Laud desired that his considerable talents should receive better opportunities for study and improvement than the obligations of constant preaching would permit. In November 1635 he had been nominated by Laud to a fellowship at All Souls College, Oxford, where, says Antony Wood, love and admiration still waited on him. He seems, however, to have spent little time there. He became chaplain to his patron the archbishop, and chaplain in ordinary to Charles I.

At Oxford, William Chillingworth was then busy with his magnum opus, The Religion of Protestants, and it is possible that through his discussions with Chillingworth Taylor may have been turned towards the liberal movement of his age. After two years in Oxford, he was presented, in March 1638, by William Juxon, Bishop of London, to the rectory of Uppingham in Rutland. There he settled down to the work of a country priest.

In the next year he married Phoebe Langsdale, by whom he had six children: William (d.1642), George (?), Richard (the last two died c.1656/7), Charles, Phoebe and Mary. In the autumn of the same year he was appointed to preach in St Mary's on the anniversary of the Gunpowder Plot, and apparently used the occasion to clear himself of a suspicion, which, however, haunted him through life, of a secret leaning to the Roman Catholic position. This suspicion seems to have arisen chiefly from his intimacy with Christopher Davenport, better known as Francis a Sancta Clara, a learned Franciscan friar who became chaplain to Queen Henrietta; but it may have been strengthened by his known connection with Laud, as well as by his ascetic habits. More serious consequences followed his attachment to the Royalist cause. The author of The Sacred Order and Offices of Episcopacy or Episcopacy Asserted against the Arians and Acephali New and Old (1642), could scarcely hope to retain his parish, which was not, however, sequestered until 1644. Taylor probably accompanied the king to Oxford. In 1643 he was presented to the rectory of Overstone, Northamptonshire, by Charles I. There he would be in close connection with his friend and patron Spencer Compton, 2nd Earl of Northampton.

Royalist prisoner
During the next fifteen years, Taylor's movements are not easily traced. He seems to have been in London during the last weeks of Charles I in 1649, from whom he is said to have received his watch and some jewels which had ornamented the ebony case in which he kept his Bible. He had been taken prisoner with other Royalists in the siege of Cardigan Castle on 4 February 1645. In 1646 he is found in partnership with two other deprived clergymen, keeping a school at Newton Hall, in the parish of Llanfihangel Aberbythych, Carmarthenshire. Here he became private chaplain to and benefited from the hospitality of Richard Vaughan, 2nd Earl of Carbery, whose mansion, Golden Grove, is immortalised in the title of Taylor's still popular manual of devotion, and whose first wife was a constant friend of Taylor. At Golden Grove Taylor wrote some of his most distinguished works. Alice, the third Lady Carbery, was the original of the Lady in John Milton's Comus. Taylor's first wife had died early in 1651. His second wife was Joanna Bridges or Brydges, said to be a natural daughter of Charles I; there is no good evidence for this. She owned a good estate, though probably impoverished by Parliamentarian exactions, at Mandinam, in Carmarthenshire. Several years following their marriage, they moved to Ireland. From time to time Taylor appears in London in the company of his friend John Evelyn, in whose Diary and correspondence his name repeatedly occurs. He was imprisoned three times: in 1645 for an injudicious preface to his Golden Grove; again in Chepstow Castle, from May to October 1655, on what charge does not appear; and a third time in the Tower in 1657, because of the indiscretion of his publisher, Richard Royston, who had decorated his Collection of Offices with a print representing Christ in the attitude of prayer.

Writings
The Rule and Exercises of Holy Living provided a manual of Christian practice, which has retained its place with devout readers. The scope of the work is described on the title page. it deals with the means and instruments of obtaining every virtue, and the remedies against every vice, and considerations serving to the resisting all temptations, together with prayers containing the whole Duty of a Christian. Holy Dying was perhaps even more popular. A very charming piece of work of a lighter kind was inspired by a question from his friend, Katherine Philipps ("the matchless Orinda"), asking How far is a dear and perfect friendship authorised by the principles of Christianity? In answer to this, he dedicated to Mrs Phillipps his Discourse of the Nature, Offices and Measures of Friendship (1657). His Ductor Dubitantium, or the Rule of Conscience … (1660) was intended to be the standard manual of casuistry and ethics for the Christian people. His works were translated into Welsh by Nathanael Jones.

Bishop in Ireland at the Restoration
He probably left Wales in 1657, and his immediate connection with Golden Grove seems to have ceased two years earlier. In 1658, through the kind offices of his friend John Evelyn, Taylor was offered a lectureship in Lisburn, County Antrim, by Edward Conway, 2nd Viscount Conway. At first, he declined a post in which the duty was to be shared with a Presbyterian – or, as he expressed it, "where a Presbyterian and myself [shall be] like Castor and Pollux, the one up the other downe" – and to which a meagre salary was attached. He was, however, induced to take it, and found in his patron's property at Portmore, on Lough Neagh, a congenial retreat.

At the Stuart Restoration, instead of being recalled to England, as he probably expected and certainly desired, he was appointed to the see of Down and Connor, to which was shortly added the additional responsibility for overviewing the adjacent diocese of Dromore. As bishop, he commissioned in 1661 the building of a new cathedral at Dromore for the Dromore diocese. He was also made a member of the privy council of Ireland and, in 1660, Vice-Chancellor of the University of Dublin. None of these positions was a sinecure.

Of the university he wrote: "I found all things in a perfect disorder ... a heap of men and boys, but no body of a college, no one member, either fellow or scholar, having any legal title to his place, but thrust in by tyranny or chance."

Accordingly, he set himself vigorously to the task of framing and enforcing regulations for the admission and conduct of members of the university, and also of establishing lectureships. His episcopal labours were still more arduous. There were, at the date of the Restoration, about seventy Presbyterian ministers in the north of Ireland, and most of these were from the west of Scotland, with a dislike for Episcopacy which distinguished the Covenanting party. No wonder that Taylor, writing to James Butler, 1st Duke of Ormonde shortly after his consecration, should have said, "I perceive myself thrown into a place of torment". His letters perhaps somewhat exaggerate the danger in which he lived, but there is no doubt that his authority was resisted and his overtures rejected.

This was Taylor's golden opportunity to show the wise toleration he had earlier advocated, but the new bishop had nothing to offer the Presbyterian clergy but the alternative of submission to episcopal ordination and jurisdiction or deprivation. Consequently, at his first visitation, he declared thirty-six churches to be vacant; and repossession was secured on his orders. At the same time, many of the gentry were apparently won over by his undoubted sincerity and devotedness as well as by his eloquence. With the Roman Catholic element of the population he was less successful. Not knowing the English language, and firmly attached to their traditional forms of worship, they were nonetheless compelled to attend a service they considered profane, conducted in a language they could not understand.

As Reginald Heber says:

"No part of the administration of Ireland by the English crown has been more extraordinary and more unfortunate than the system pursued for the introduction of the Reformed religion. At the instance of the Irish bishops Taylor undertook his last great work, the Dissuasive from Popery (in two parts, 1664 and 1667), but, as he himself seemed partly conscious, he might have more effectually gained his end by adopting the methods of Ussher and William Bedell, and inducing his clergy to acquire the Irish language."

During this period, he was married a second time to Joanna Brydges, supposedly a natural daughter of Charles I. From this marriage, two daughters were born: Mary, who went on to marry Archbishop Francis Marsh and had issue and Joanna, who married Edward Harrison, MP for Lisburn, and had issue. From his father-in-law, Marsh inherited a silver watch, said to have been a gift from Charles I; this watch remained in the family of his great-grandson, Francis Marsh, barrister-at-law. Taylor died at Lisburn on 13 August 1667. He was buried at Dromore Cathedral where an apsidal chancel was built in 1870 over the crypt where he was laid to rest.

Jeremy Taylor is honoured in the Church of England, the Anglican Church of Canada, Scottish Episcopal Church, Anglican Church of Australia and in the Episcopal Church of the United States on 13 August.

<!--This material is sourced from the Encyclopaedia Britannica, but represents an individual assessment by the anonymous author and does not seem to be encyclopedic.

Thought
Taylor's fame has been maintained by the popularity of his sermons and devotional writings rather than by his influence as a theologian or his importance as an ecclesiastic. His mind was neither scientific nor speculative, and he was attracted rather to questions of casuistry than to the problems of pure theology. His wide reading and capacious memory enabled him to carry in his mind the materials of a sound historical theology, but these materials were unsifted by criticism. His immense learning served him rather as a storehouse of illustrations, or as an armoury out of which he could choose the fittest weapon for discomfiting an opponent, than as a quarry furnishing him with material for building up a completely designed and enduring edifice of systematised truth. Indeed, he had very limited faith in the human mind as an instrument of truth. Theology, he says, is rather a divine life than a divine knowledge.

His great plea for toleration is based on the impossibility of erecting theology into a demonstrable science. It is impossible all should be of one mind. And what is impossible to be done is not necessary it should be done. Differences of opinion there must be; but heresy is not an error of the understanding but an error of the will. He would submit all minor questions to the reason of the individual member, but he set certain limits to toleration, excluding whatsoever is against the foundation of faith, or contrary to good life and the laws of obedience, or destructive to human society, and the public and just interests of bodies politic. Peace, he thought, might be made if men would not call all opinions by the name of religion, and superstructures by the name of fundamental articles. Of the propositions of sectarian theologians he said that confidence was the first, and the second, and the third part.

Of a genuine poetic temperament, fervid and mobile in feeling, and of a prolific fancy, he had also the sense and wit that come of varied contact with men. All his gifts were made available for influencing other men by his easy command of a style rarely matched in dignity and colour. With all the majesty and stately elaboration and musical rhythm of Milton's finest prose, Taylor's style is relieved and brightened by an astonishing variety of felicitous illustrations, ranging from the most homely and terse to the most dignified and elaborate. His sermons especially abound in quotations and allusions, which have the air of spontaneously suggesting themselves, but which must sometimes have baffled his hearers. This seeming pedantry is, however, atoned for by the clear practical aim of his sermons, the noble ideal he keeps before his hearers, and the skill with which he handles spiritual experience and urges incentives to virtue.

Literary style and influence
Taylor is best known as a prose stylist; his chief fame is the result of his twin devotional manual, Holy Living and Holy Dying. (The Rule and Exercises of Holy Living, 1650 and The Rule and Exercises of Holy Dying, 1651). These books were favourites of John Wesley, and admired for their prose style by Samuel Taylor Coleridge, William Hazlitt, and Thomas de Quincey. They are marked by solemn but vivid rhetoric, elaborate periodic sentences, and careful attention to the music and rhythms of words:

As our life is very short, so it is very miserable; and therefore it is well that it is short. God, in pity to mankind, lest his burden should be insupportable and his nature an intolerable load, hath reduced our state of misery to an abbreviature; and the greater our misery is, the less while it is like to last; the sorrows of a man's spirit being like ponderous weights, which by the greatness of their burden make a swifter motion, and descend into the grave to rest and ease our wearied limbs; for then only we shall sleep quietly, when those fetters are knocked off, which not only bound our souls in prison, but also ate the flesh until the very bones opened the secret garments of their cartilages, discovering their nakedness and sorrow.
 * —From Rules and Exercises of Holy Dying

end of EB1911 omitted material -->

Family
Jeremy Taylor is said to have been a lineal descendant of Rowland Taylor, but the assertion has not been proved. Through his daughter, Mary, who married Archbishop Francis Marsh, he had numerous descendants.

Principal publications

 * A Discourse of the Liberty of Prophesying (1646), a famous plea for toleration published decades before John Locke's Letters Concerning Toleration.
 * Apology for Authorised and Set Forms of Liturgy against the Pretence of the Spirit (1649)
 * Great Exemplar … a History of … Jesus Christ (1649), inspired, its author tells us, by his earlier intercourse with the earl of Northampton
 * The Rule and Exercises of Holy Living (1650) | eBook
 * The Rule and Exercises of Holy Dying (1651) | eBook
 * Twenty-seven Sermons Preached at Golden Grove, for the Summer Half-year … (1651)
 * Twenty-five Sermons Preached at Golden Grove, for the Winter Half-year … (1653)
 * Clerus Domini: or, A Discourse of the Divine Institution, Necessity, Sacrednesse, and Separation of the Office Ministerial (1651)
 * The Real Presence and Spirituall of Christ in the Blessed Statement Proved Against the Doctrine of Transubstantiation. (1654)
 * Golden Grove; or a Manuall of Daily Prayers and Letanies [sic] … (1655)
 * Unum Necessarium (1655), on the doctrine of repentance, perceived Pelagianism gave great offence to Presbyterians.
 * Discourse of the Nature, Offices and Measures of Friendship (1657)
 * Ductor Dubitantium, or the Rule of Conscience … (1660)
 * The Worthy Communicant; or a Discourse of the Nature, Effects, and Blessings consequent to the worthy receiving of the Lords Supper … (1660)
 * Contemplations of the state of man in this life and in that which is to come (1684)

Secondary literature and posthumous editions

 * Antoine, Sister Mary Salome. The Rhetoric of Jeremy Taylor's Prose: Ornament of the Sunday Sermons. PhD Dissertation. Washington, D.C.: Catholic University of America Press, 1946.
 * Bonney, Henry Kaye. The Life of ... Jeremy Taylor ... Lord Bishop of Down, Connor, and Dromore. London: T. Cadell & W. Davies, 1815.
 * Brown, W. J. Jeremy Taylor. London: Society for Promoting Christian Knowledge, 1925.
 * Carroll, Thomas K., ed. Jeremy Taylor: Selected Works. Preface by John Booty. The Classics of Western Spirituality Series. New York City: Paulist Press, 1990.
 * de Ricci Albrecht, Sister Mary Catherine. The Exemplum in the Sermons of Jeremy Taylor. M.A. dissertation. District of Columbia: Catholic University of America, 1947.
 * Duyckinck, George L. The Life of Jeremy Taylor, Bishop of Down, Connor, and Dromore. New York: 1860.
 * Gosse, Edmund. Jeremy Taylor. The English Men of Letters Series. London: Macmillan & Co., Ltd., 1904.
 * Grosart, A.B., ed. The Poems and Verse Translations of Jeremy Taylor. Fuller Worthies’ Library. 1870.
 * Gathorne-Hardy, Robert. A Bibliography of the Writings of Jeremy Taylor to 1700. Dekalb, IL: Northern Illinois University, 1971.
 * Heber, Reginald, ed. The Whole Works of the Right Rev. Jeremy Taylor […], with a Life of the Author, and a Critical Examination of His Writings, in Ten Volumes. Revised and corrected by Charles Page Eden. London: Longman, Green, and Longmans, 1848.
 * Herndon, S. Jeremy Taylor's Use of the Bible. PhD. dissertation. New York: New York University, 1949.
 * Hughes, H. Trevor. The Piety of Jeremy Taylor. London: Macmillan, 1960.
 * Huntley, Frank L. Jeremy Taylor and the Great Rebellion: A Study of His Mind and Temper in Controversy. Ann Arbor: University of Michigan Press, 1970.
 * Jackson, Robert S. The Meditative Life of Christ: A Study of the Background and Structure of Jeremy Taylor's "The Great Exemplar." PhD. dissertation. Ann Arbor: U of MI, 1959.
 * Peterson, Raymond A. The Theology of Jeremy Taylor: An Investigation of the Temper of Caroline Anglicanism. PhD. dissertation. New York: Union Theological Seminary, 1961.
 * Stranks, Charles James. The Life and Writings of Jeremy Taylor. London: Church Historical Society, 1952.
 * Streatfield, K.M. A Critical Edition of Six Occasional Sermons by Jeremy Taylor (1613–1667). PhD Dissertation. University of Edinburgh, 1988.
 * Wheeldon, John. The Life of Bishop Taylor, and the Pure Spirit of his Writings, Extracted and Exhibited for General Benefit. London: George Bigg, 1793.
 * Williamson, Hugh Ross. Jeremy Taylor. London: Dennis Dobson Ltd., 1952.
 * Worley, G. Jeremy Taylor: A Sketch of His Life & Times, with a Popular Exposition of His Works. London: Longmans, Green, & Co., 1904.

title: Jeremy Taylor