Caroline Fontana

Anaïs Kirsch

Carole Seidel

Léa Marpeaux

Christian Darne

Laurent Gaté

Aurélie Rémy

Yves Guichard
email: yves.guichard@inrs.fr

Aurélie Remy

In vitro cell transformation induced by synthetic amorphous silica nanoparticles

Keywords: Synthetic amorphous silica, Nanomaterial, Cell transformation assay, Bhas 42

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The worldwide production of synthetic amorphous silica nanoparticles (SAS) was estimated to be one million tons per year in 2006. These nanomaterials are used in many industrial applications such as chemistry, pharmaceuticals, cosmetics and foods [1]. Two main forms of SAS are produced in industries: precipitated SAS, obtained by acidification of sodium silicate under wet conditions; and pyrogenic SAS, obtained by hydrolysis of SiCl 4 at high temperature [START_REF] Fruijtier-Polloth | The toxicological mode of action and the safety of synthetic amorphous silicaa nanostructured material[END_REF]. Because of their amorphous crystal phase, SAS belong to the group of silica particles that are not classifiable as to their carcinogenicity to humans (Group 3), according to the International Agency for Research of Cancer (IARC) [START_REF] Iarc | Silica, some silicates, coal dust and para-Aramid fibrils[END_REF]. However, there is a lack of information about the cancer risk associated with SAS exposure. A full assessment of the hazards and risks for producers and users of these nanomaterials is therefore necessary.

Several in vitro and in vivo studies have reported cytotoxic and inflammatory effects induced by SAS of various origins and characteristics [START_REF] Fruijtier-Polloth | The toxicological mode of action and the safety of synthetic amorphous silicaa nanostructured material[END_REF][START_REF] Napierska | The nanosilica hazard: another variable entity[END_REF]. Data on the genotoxic properties of SAS, mostly assessed by the comet assay for detecting DNA damage or by the micronucleus assay for chromosomal damage, are however less clear. Generally, positive responses in genotoxicity assays are reported more frequently in vitro [START_REF] Barnes | Reproducible comet assay of amorphous silica nanoparticles detects no genotoxicity[END_REF][START_REF] Mu | Mechanism of cellular uptake of genotoxic silica nanoparticles[END_REF][START_REF] Gerloff | Cytotoxicity and oxidative DNA damage by nanoparticles in human intestinal Caco-2 Cells[END_REF][START_REF] Tavares | Genotoxicity evaluation of nanosized titanium dioxide, synthetic amorphous silica and multi-walled carbon nanotubes in human lymphocytes[END_REF][START_REF] Guichard | Cytotoxic and genotoxic evaluation of different synthetic amorphous silica nanomaterials in the V79 cell line[END_REF] than in vivo [START_REF] Johnston | Pulmonary chemokine and mutagenic responses in rats after subchronic inhalation of amorphous and crystalline silica[END_REF][START_REF] Guichard | Genotoxicity of synthetic amorphous silica nanoparticles in rats following short-term exposure. Part 2: intratracheal instillation and intravenous injection[END_REF][START_REF] Tarantini | Genotoxicity of synthetic amorphous silica nanoparticles in rats following short-term exposure. Part 1: oral route[END_REF]. However, in vitro results remain quite variable and depend on the physical and chemical characteristics of the SAS and the cell type used. On the other hand, little is known about the carcinogenic potential of SAS. To our knowledge, only one long-term carcinogenesis study in rats and one in vitro study using a cell transformation assay (CTA) have been published to date [START_REF] Pott | Carcinogenicity study with nineteen granular dusts in rats[END_REF][START_REF] Uboldi | Amorphous silica nanoparticles do not induce cytotoxicity, cell transformation or genotoxicity in Balb/3T3 mouse fibroblasts[END_REF].

The aim of the present study was to investigate the capacity of four different SAS (NM-200, NM-201, NM-202 and NM-203) to induce cell transformation. We therefore used the in vitro Bhas 42 CTA, which was recently the subject of a guidance document produced by the Organization for Economic Cooperation and Development (OECD) [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF].

The principle of this assay is the following [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF][START_REF] Sasaki | Transformation assay in Bhas 42 cells: a model using initiated cells to study mechanisms of carcinogenesis and predict carcinogenic potential of chemicals[END_REF]: the transgenic Bhas 42 cell line, established from Balb/c 3T3 cells, contains several copies of the v-Ha-ras gene. Similar to the parental Balb/c 3T3 cells, untransformed Bhas 42 cells grow to confluence forming a density-dependent contact-inhibited monolayer. However, after exposure to transforming stimuli, such cells become morphologically altered and form discrete colonies, referred to as transformed foci, atop the confluent monolayer. Interestingly, because they contain an activated oncogene, Bhas 42 cells can respond to tumor-promoter substances without undergoing preliminary treatment with an initiating substance, in contrast with other twostage cell transformation models. Furthermore, tumor-initiator substances can also transform Bhas 42 cells, for example, by inducing mutation of transformation-related genes (oncogenes and/or tumor suppressor genes). Based on this, the Bhas 42 CTA two transformation assays were performed in parallel (Figure 1). In the initiation assay, cells are seeded at low density and then treated once during the cell-growth phase (from Day 1 to Day 4). Cells are allowed to replicate and are then fixed and stained on Day 21 after plating for evaluation of the resulting transformed foci. These seeding and treatment conditions enable DNA damage and mutation fixation and favor the initiating activity of chemicals. In the promotion assay, cells are seeded at higher density. A first treatment is started at sub-confluence and is then repeated twice during a 10 day period (from Day 4 to Day 14). As in the initiation assay, the cells are then fixed and stained on Day 21 after plating. These conditions allow cell-to-cell communication and favor the promoting activity of chemicals. Thus, Bhas 42 cells can be used not only for the screening of chemicals with transforming activity but also for studying mechanisms of cell transformation.

In the present study, the suitability of the Bhas 42 CTA for particle-type substances was assessed using crystalline (quartz Min-U-Sil 5) and amorphous (diatomaceous earth) silica microparticles that have been previously evaluated for their transforming activity in the Syrian hamster embryonic (SHE) CTA [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF][START_REF] Darne | In vitro comet and micronucleus assays do not predict morphological transforming effects of silica particles in Syrian Hamster Embryo cells[END_REF]. previous study [START_REF] Rasmussen | Characterisation and physico-chemical properties. JRC repository: NM-series of representative manufactured banomaterials[END_REF] (Table 1). Min-U-Sil and DE batches have also been characterised previously [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF].

Materials and methods

Cell culture

Bhas 42 cells were established from the v-Ha-ras-transfected BALB/c 3T3 cells by Sasaki et al. [START_REF] Sasaki | Isolation and characterization of ras-transfected BALB/3T3 clone showing morphological transformation by 12-O-tetradecanoyl-phorbol-13-acetate[END_REF]. Frozen Bhas 42 cells at passage 19 were obtained from Harlan Laboratories (Rossdorf, Germany). In order to generate a large batch of cells from the same pool, the cells were cultivated in Eagle's minimum essential medium (Invitrogen, France), supplemented with 1% antibiotic (Streptomycin and Penicillin, Invitrogen) and 10% fetal bovine serum (FBS, Dutscher, France) (M10F) in a humidified 5% CO 2 incubator at 37 °C. When the cultures reached about 70% confluence, they were subcultured using 0.25% trypsin (Invitrogen).

Within two passages after thawing, the cultured cells were suspended at 5×10 5 cells/mL in fresh M10F containing 5% dimethyl sulfoxide (DMSO, Sigma Aldrich, France), frozen in 0.5 mL volumes at -80 °C and stored in liquid nitrogen. All CTA performed in this study used this frozen batch of cells (passage 22), which was confirmed to be free from mycoplasma contamination. Dulbecco's modified Eagle's medium/Ham's F12 (Invitrogen) supplemented with 1% antibiotic and 5% FBS (DF5F) was used for CTA. The batch of FBS used throughout this study was pre-screened to produce a minimal number of transformed foci in the Bhas 42 cells treated with the solvent control, DMSO, and a substantially elevated number of transformed foci in the cells treated with positive controls, 3-methylcholanthrene (MCA, Sigma Aldrich) at 1 µg/mL in the initiation assay or 12-O-tetradecanoylphorbol-13-acetate (TPA, Sigma Aldrich) at 0.05 µg/mL in the promotion assay [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF].

Cell treatment with particles

Particle powders were mixed in a 20 mL glass vial with sterile de-ionized water. The concentration of particles was adjusted to 4.8 mg/mL for Min-U-Sil and 19.2 mg/mL for DE and SAS. The tube was placed in the cup horn of a 400 W Branson Sonifier S-450D (Branson Ultrasonics Corp, Danbury, CT) containing ice-cold water and then sonicated for 5 min at 10% amplitude. Particle size distribution was analysed by dynamic light scattering (DLS), using a Malvern Nano ZS (Malvern Inc., UK), immediately after preparation. For cell treatment, serial dilutions were prepared in sterile de-ionized water from the sonicated suspension. An appropriate volume of each of these preparations was diluted ten times for Min-U-Sil and fifty times for DE and SAS in DF5F (see section 2.4.). Final particle concentrations were expressed as mass per cell culture surface area. The concentrations were tested ranged from 5 µg/cm 2 to 100 µg/cm 2 for Min-U-Sil and from 2 µg/cm 2 to 80 µg/cm 2 for DE and SAS.

4. CTA and concurrent cell growth assays

The CTA and concurrent cell growth assays in their six-well format were performed in accordance with the recommendations of the OECD [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF]. The protocol is based on an initiation assay and a promotion assay and is illustrated in Figure 1. One tube of the frozen Bhas 42 cells (passage 22) was thawed and grown in M10F up to about 70% confluence and then subcultured in DF5F to reach about 70% confluence again. The cells were trypsinised and suspended either at a density of 2000 cells/ml for seeding in 2 mL volume intended for the initiation assay (4000 cells/well, Day 0), or at a density of 7000 cells/ml for seeding in 2 mL volume intended for the promotion assay (14,000 cells/well, Day 0). In the initiation assay, the cells were treated 24 h after seeding (Day 1) by the addition of a particle suspension, MCA solution (1mg/mL), or the vehicles (sterile de-ionized water or DMSO). The treatment was continued for 72 h (Day 4). Next, all treatment media were removed, and the cells were post-cultivated in fresh medium until Day 21, receiving medium exchanges on Day 7, Day 10 and Day 14. In the promotion assay, the cells were cultured after seeding for four days without a medium exchange. In the regular CTA promotion assay (Figure 1), the culture medium was replaced on Day 4, Day 7, and Day 10 with a fresh medium containing particle suspensions, TPA solution (0.05 mg/ml) or the vehicles, and the treatment in the promotion phase was continued until Day 14 (for a total of 10 days). Observations of the cell cultures under a microscope showed that the majority of the tested particles were removed after each culture medium change. The cells were then cultured in the normal medium without the test chemical for one week until Day 21. A modified CTA promotion assay was also performed, in which cells were treated for three days only once, on Day 4. The cells were then postcultivated in fresh medium until Day 21, receiving medium exchanges on Day 7, Day 10 and Day 14. For all CTA, the cells were fixed with ethanol and stained with a 5 % Giemsa solution (Fisher, France) on Day 21 and the transformed foci were evaluated according to the morphological criteria recommended by OECD [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF]. For each experiment, the mean of the number of transformed foci per treatment group was determined from the six replicate wells.

Statistical analysis was performed on results obtained from three independent experiments. CTA results were considered positive when there were two or more sequential concentrations that induced statistically significant increases in the number of transformed foci, and negative when no concentration showed a statistically significant increase in foci.

Cell growth assays were performed on Day 7. The colorimetric method (with crystal violet) recommended by OECD [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF] could not be applied to cultures treated with nano-or microparticles because the presence of particle agglomerates in wells interfered with the measure of the optical density. The Trypan-Blue exclusion method was used as alternative method. In a single well dedicated to the growth assay, cells were recovered using 0.25% trypsin, stained with 0.1% Trypan-Blue solution (Sigma-Aldrich), and then counted under a microscope using a Nageotte hemocytometer. The relative cell growth of cultures treated with particles was calculated as follows:

Relative cell growth (%) = [(Treatment)/(Control)] x 100 where "Treatment" and "Control" refer to the number of viable cells counted in each treatment group and the vehicle control group, respectively. The concentration of particles used to assess their transformation activity was selected according to the results of two independent cell growth assays. These concentrations ranged from little or no toxicity to the highest acceptable level of toxicity (less than 20% survival compared to the vehicle control) [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF]. The maximum particle concentration did not exceed 80 or 100 µg/cm 2 in any case because the higher concentrations could have smothered cell cultures.

5. Statistical analysis

Data are given as a mean ± standard deviation (mean ± SD). Statistical analyses were performed using Stata 14.0 software. A logarithmic transformation was applied to the data.

The mixed linear regression model was used to test the "dose" fixed effect, including a random effect "experience". A multiple comparisons post hoc test (with Bonferroni correction) was applied to test the difference between control and different levels of doses. The statistical significant level was set to 5%.

Results

Transforming effect of MCA and TPA

MCA (1 µg/mL) and TPA (0.05 µg/mL) were used as positive controls in all cell transformation assays performed with silica micro-and nanoparticles. In this study, the average number of transformed foci per well induced by MCA in initiation assays was 13.2 ± 3.1, versus 3.2 ± 1.6 transformed foci in the control (0.1% DMSO), and the average number of transformed foci per well induced by TPA in promotion assays was 14.9 ± 3.0, versus 5.7

2. Transforming effect of Min-U-Sil and DE

Bhas 42 CTA results are shown in Figure 2 dependant manner in the initiation assay, with a 50% inhibitory concentration (IC50) of around 20 µg/cm 2 . In contrast, the IC50 was not reached in the promotion assay. DE induced a decrease in cell proliferation in a concentration-dependant manner in the initiation assay with an IC50 of around 20 µg/cm 2 . In the promotion assay the IC50 was reached at 80 µg/cm 2 .

In the initiation assay, Min-U-Sil treatment did not increase the number of transformed foci compared to the control. In the promotion conditions, however, Min-U-Sil treatment induced cell transformation, producing a bell-shaped response with a lowest observed adverse effect level (LOAEL) at 15 µg/cm 2 . In contrast, DE did not show any transforming activity at any of the concentrations tested in the initiation and promotion assays. On the contrary, the number of transformed foci decreased with the increase of particle concentrations.

2. Transforming effect of SAS

Bhas 42 CTA results are shown in Figure 3 and images of representative CTA are shown in Figure S2. With the exception of NM-200 treatment in the initiation assay, all SAS decreased cell proliferation in a concentration-dependant manner in both the initiation and promotion assay. In the initiation assay, the IC50 were around 40 µg/cm 2 for NM-201 and NM-202, and around 20 µg/cm 2 for NM-203. In the promotion assay, NM-202 and NM-203 showed the lowest IC50, at around 20 µg/cm 2 , while for NM-200 and NM-201, the IC50 was reached at 80 µg/cm 2 .

In the initiation assay, none of the SAS substantially increased the number of transformed foci compared to the control. In contrast, in the promotion assay, the four SAS significantly increased the number of transformed foci at at least two sequential concentrations. A bellshaped pattern of cell transformation was observed for all SAS. The LOAEL for cell transformation was observed at 2 µg/cm 2 for NM-203, 10 µg/cm 2 for NM-200 and NM-202 and 20 µg/cm 2 for NM-201.

Transforming effect of Min-U-Sil and NM-203 in the promotion assay performed with a single treatment

The Transforming activity of Min-U-Sil and NM-203 were tested in a modified promotion assay, in which Bhas 42 cells were treated only on Day 4 (Figure 4). Under these experimental conditions, both treatments increased the average number of transformed foci compared to the control, but at a lower magnitude than under the conditions of the regular promotion assay (with three sets of treatment as shown in Figure 2 and3). The LOAEL for cell transformation was observed at 20 µg/cm 2 for Min-U-Sil and at 5 µg/cm 2 for NM-203.

Discussion

Few data are available on the carcinogenic potential of SAS. To our knowledge, only one in vivo carcinogenesis study is reported in the literature. This study showed that five repeated intratracheal instillations of SAS in rats (3 mg per instillation) increased the incidence of lung tumours [START_REF] Pott | Carcinogenicity study with nineteen granular dusts in rats[END_REF]. Interestingly, the characteristics of the SAS used in the study of [START_REF] Pott | Carcinogenicity study with nineteen granular dusts in rats[END_REF] were very similar to NM-203. On the other hand, an in vitro cell transformation assay using Balb/c 3T3 cells showed no transforming activities for different SAS, including NM-200 and NM-203 [START_REF] Uboldi | Amorphous silica nanoparticles do not induce cytotoxicity, cell transformation or genotoxicity in Balb/3T3 mouse fibroblasts[END_REF].

In the present study, we used an in vitro model of cell transformation; the transgenic Bhas 42 cells that are particularly sensitive to tumor-promoter substances but also being sensitive to tumor-initiator substances [START_REF] Sasaki | Transformation assay in Bhas 42 cells: a model using initiated cells to study mechanisms of carcinogenesis and predict carcinogenic potential of chemicals[END_REF]. A validation study covering 98 chemicals established an 84% specificity of the Bhas 42 CTA towards carcinogens and non-carcinogens [START_REF] Sakai | A Bhas 42 cell transformation assay on 98 chemicals: the characteristics and performance for the prediction of chemical carcinogenicity[END_REF].

In this study, the numbers of transformed foci obtained in control culture and in treated culture (MCA in the initiation assay and TPA in the promotion assay) were in agreement with those previously published [START_REF] Sakai | A Bhas 42 cell transformation assay on 98 chemicals: the characteristics and performance for the prediction of chemical carcinogenicity[END_REF][START_REF] Sakai | An international validation study of a Bhas 42 cell transformation assay for the prediction of chemical carcinogenicity[END_REF]. At the beginning of our study, the Bhas 42 CTA had not yet been tested with particle-type substances. Thus, we first submitted the assay to a known carcinogenic silica (Min-U-Sil) and to an amorphous silica (DE) so far not classified as carcinogenic [START_REF] Iarc | Silica, some silicates, coal dust and para-Aramid fibrils[END_REF][START_REF]Metals, Fibres and Dusts[END_REF]. As expected, the Bhas 42 CTA was also able to detect the transforming activity of Min-U-Sil, but interestingly only in the promotion assay. DE treatment did not induce cell transformation, but decreased the number of transformed foci in the initiation and promotion assays. Other non-carcinogen chemicals such as L-ascorbic acid or caffeine have been found to lower the number of spontaneous transformed Bhas 42 foci [START_REF] Sakai | A Bhas 42 cell transformation assay on 98 chemicals: the characteristics and performance for the prediction of chemical carcinogenicity[END_REF].. Nevertheless, the fact that DE did not induce cell transformation in the Bhas 42 assay strongly supports its specificity towards particle-type substances.

In this study, we focused on the transforming potential of representatives of pyrogenic or precipitated SAS produced in industry. Interestingly, the same batches of SAS were evaluated for their toxicological effect in previous in vivo or in vitro studies. All four SAS were able to cause lung inflammation in rats by intratracheal instillation, but did not induce any DNA damage, neither in the lung nor in other secondary organs [START_REF] Guichard | Genotoxicity of synthetic amorphous silica nanoparticles in rats following short-term exposure. Part 2: intratracheal instillation and intravenous injection[END_REF]. Similarly, none of them induced DNA damage in different tissues of rats exposed by gavage. However, a weak genotoxic effect was detected by the micronucleus assay in colon cells from rats treated by gavage with the two pyrogenic silica nanoparticles (NM-202 and NM-203) [START_REF] Tarantini | Genotoxicity of synthetic amorphous silica nanoparticles in rats following short-term exposure. Part 1: oral route[END_REF]. On the other hand, the four silica nanoparticles that we used were unable to induce in vitro micronuclei in human lymphocytes [START_REF] Tavares | Genotoxicity evaluation of nanosized titanium dioxide, synthetic amorphous silica and multi-walled carbon nanotubes in human lymphocytes[END_REF], and micronucleus assays were negative in Balb/c 3T3 exposed to NM-200 or NM-203 [START_REF] Uboldi | Amorphous silica nanoparticles do not induce cytotoxicity, cell transformation or genotoxicity in Balb/3T3 mouse fibroblasts[END_REF].

To our knowledge, our study is the first study to have shown the in vitro transforming activity of SAS. Cell proliferation assays indicated higher levels of cytotoxicity for the pyrogenic SAS (NM-202 and NM-203) compared to the precipitated SAS. Interestingly, Tavares et al. did not observe any cytotoxic activity after 30h treatment in human lymphocytes with any of the four SAS [START_REF] Tavares | Genotoxicity evaluation of nanosized titanium dioxide, synthetic amorphous silica and multi-walled carbon nanotubes in human lymphocytes[END_REF]. In our study, all SAS exhibited a bell-shaped-patterned transforming effect in the promotion assay. The decrease in cell transformation observed at the highest concentration of SAS can probably be explained by more pronounced growth inhibition after 21 days of culture, which is emphasised by the application of three sets of treatment. This type of pattern has also been observed for soluble transforming substances in the Bhas 42 CTA [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF][START_REF] Sakai | A Bhas 42 cell transformation assay on 98 chemicals: the characteristics and performance for the prediction of chemical carcinogenicity[END_REF]. Of the four SAS tested in the Bhas 42 CTA, NM-203, having induced cell transformation at the lowest concentration (2 µg/cm 2 ), appeared to have the highest transforming potency.

Rasmussen et al. reported that NM 203 particle aggregates possessed a particularly complex fractal structure that is associated with high production of oxygen species in aqueous solution [START_REF] Rasmussen | Characterisation and physico-chemical properties. JRC repository: NM-series of representative manufactured banomaterials[END_REF]. This observation could explain the differential transforming effect of NM-203 compared to the other SAS.

Like with Min-U-Sil, the transforming effect of the four SAS was only detected under experimental conditions of promotion. On the other hand, the absence of any transforming effects of SAS in the initiation assay is consistent with the lack of genotoxicity of NM-200 or NM-203 previously shown in the Balb/c 3T3 cells [START_REF] Uboldi | Amorphous silica nanoparticles do not induce cytotoxicity, cell transformation or genotoxicity in Balb/3T3 mouse fibroblasts[END_REF]. Together, these observations suggest that the mode of action of SAS is similar to that of tumor-promoter agents. In order to test this hypothesis, we modified the promotion assay, performing only a single treatment of Min-U-Sil or NM-203 instead of the usual three sets of treatment. We observed that Min-U-Sil and NM-203 were still capable of inducing cell transformation, but at a lower magnitude than in the original procedure. This demonstrated the importance of the role of cell confluence at the time of treatment with either type of silica particles; cell confluence was much higher under the promotion conditions than it was under the initiation conditions, where no transformation effect was detected. As the authors of the Bhas 42 CTA point out, cell confluence favours intercellular communication, which is recognised as a key factor of promotion mechanisms in carcinogenesis processes [START_REF] Oecd | Guidance Document On the In Vitro Bhas 42 Cell Transformation Assay Series on Testing & Assessment[END_REF]. The experiment also showed that in the context of the original promotion assay, the first of the three treatments with Min-U-Sil or NM-203 was sufficient to trigger a cell transformation process. When a single treatment is performed, the promotion mechanism is probably maintained by the persistence of particle uptake throughout the assay. Some studies have shown that high levels of SAS airborne particles can be found in situations of occupational exposure [START_REF] Oh | Comparison of Nanoparticle Exposures Between Fumed and Sol-gel Nanosilica Manufacturing Facilities[END_REF][START_REF] Kim | Assessment of Nanoparticle Exposure in Nanosilica Handling Process: Including Characteristics of Nanoparticles Leaking from a Vacuum Cleaner[END_REF]. However, based on an estimation derived from human tolerable levels, Murugadoss et al. concluded that relevant SAS concentrations for in vitro assays should not exceed 384 µg/cm 2 [START_REF] Murugadoss | Toxicology of silica nanoparticles: an update[END_REF]. Although it is generally difficult to extrapolate the in vitro dose to the in vivo status, SAS concentrations used in this study (ranging from 2 µg/cm 2 to 80 µg/cm 2 ) could be considered as physiologically relevant.

In conclusion, this study has shown, for the first time, the transforming potential of different SAS. We employed a promising model of cell transformation, the Bhas 42 CTA, which was designed for the detection of tumor-promoter agents. Further studies are now needed to understand, at the molecular level, the mechanisms of promotion involved in the transforming effect of SAS, such as epigenetic events. With the exception of DLS data, detailed physical and chemical characterizations are available in [START_REF] Rasmussen | Characterisation and physico-chemical properties. JRC repository: NM-series of representative manufactured banomaterials[END_REF] for SAS and [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF] for Min-U-Sil and DE.

b Primary size was determined by transmission electron microscopy.

2. 1

 1 . Particles SAS (NM-200, NM-201, NM-202, and NM-203) were obtained from the Joint Research Centre nanomaterials repository (Ispra, Italy). Min-U-Sil 5 (Min-U-Sil) was obtained from US Silica Co. (Berkeley). Diatomaceous earth (DE) was obtained from CECA (Paris, France). A thorough physical and chemical characterisation of SAS batches was conducted in a

± 2 .

 2 0 transformed foci in the control (0.1% DMSO). Images of transformed foci obtained from CTA are shown in Figure S.1.

  and photographs of representative CTA are shown in Figure S.2. Min-U-Sil induced a decrease in cell proliferation in a concentration-

c

  Impurities were determined by energy dispersive spectrometry for SAS and by inductively coupled plasma spectrometry for Min-U-Sil and DE.dThe BET (Brunauer-Emmet-Teller) specific surface area (SSA) was determined by nitrogen adsorption. e Dynamic light scattering (DLS) was performed using a Nanozetasizer (Malvern). Samples were dispersed in water as described in the Materials and Methods section. Values represent the mean ± standard deviation of three DLS analyses.

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Results of initiation and promotion Bhas 42 CTA (histogram, n = 6, * p < 0.05) and cell growth assays (curve, # p < 0.05) on Min-U-Sil and diatomaceous earth (DE). C: control (treatment with 10% H 2 O for Min-U-Sil or 2% H 2 O for DE). Results are expressed as the mean ± SD of three independent experiments for CTA or two independent experiments for cell growth assays.

Figure 3 :

 3 Figure 3: Results of initiation and promotion Bhas 42 CTA (histogram, n = 6, *p < 0.05) and cell growth assays (curve, # p < 0.05) on SAS. C: control (treatment with 2% H 2 O). SAS: synthetic amorphous silica nanoparticles. Results are expressed as the mean ± SD of three independent experiments for CTA or two independent experiments for cell growth assays.

Figure 4 :

 4 Figure 4: Results of the modified promotion Bhas 42 CTA (treatment on Day 4 only) (histogram, n = 6, *p < 0.05) on Min-U-Sil and NM203. C: control (treatment with 10% H 2 O for Min-U-Sil or 2% H 2 O for NM-203). Results are expressed as the mean ± SD of three independent experiments for CTA.

Figure S. 1 :

 1 Figure S.1: Representative cell cultures obtained on Day 21 from CTA performed with 3methylcholanthrene (MCA) in the initiation assay or 12-O-tetradecanoylphorbol-13-acetate (TPA) in the promotion assay. Treatments with 0.1% dimethyl sulfoxide (DMSO) were used as the negative control.

Figure S. 2 :

 2 Figure S.2: Representative cell cultures obtained on Day 21 from CTA performed with DE, Min-U-Sil or SAS in the initiation assay and in the promotion assay. Treatments with 2% H 2 O were used as negative controls in all CTA except for Min-U-Sil, for which the negative control was 10% H 2 O (not shown).

Table 1 :

 1 Physical characteristics of silica particles used in this study a

  Not determined, data obtained by DLS analysis did not meet the quality criteria.

							DLS e
	Silica a	Origin	Main phase	Primary	Impurities	BET SSA	Z-average in	Polydispersity
				particle	(wt%) c	(m 2 /g) d	Water (nm)	index
				size b
	NM-200	Precipitated	Amorphous	18.3 ± 4.5 nm	Al (0.87); Na (0.88); S (0.46)	189	ND f	ND
	NM-201	Precipitated	Amorphous	18.0 ± 1.0 nm	Al (0.74); Na (0.44); S (0.46)	140	252.9 ± 11.3	0.583 ± 0.019
	NM-202	Pyrogenic	Amorphous	17.7 ± 2.5 nm	Al (0.45); Ca (0.18)	204	232.8 ± 7.3	0.441 ± 0.012
	NM-203	Pyrogenic	Amorphous	24.7 ± 17.7 nm	Al (0.43); S (0.04)	204	318.7 ± 4.9	0.423 ± 0.026
	Min-U-Sil	Natural	Crystalline	1.33 ± 0.85 µm	Al (0.16); Fe (0.05); Ca (0.04); Mg (0.01)	5	1385.7 ± 377.0	0.318 ± 0.130
	DE	Natural	Amorphous 1.35 ± 1.48 µm Al (1.4); Fe (1.3); Ca (2.

f


A theory of functions of several variables applied to square matrices

Laurent Veysseire

In this paper, we give one possible definition for functions of several variables applied to endomorphisms of finite dimensional C-vector spaces. This definition is consistent with the usual notion of a function of a square matrix. The fact that with this definition, f ⊗ (A, B) is not a square matrix of size n anymore when A and B are square matrices of size n (well, except in the trivial case where n = 1) can seem weird and unsatisfying, but this objects naturally appear when one differentiates a smooth function of a matrix.

This work was supported by the TECHNION, Israel Institute of Tecnology, Haifa, Israel.

definitions and notations

As for defining functions in one variable applied to a single matrix (as done in [START_REF] Gantmacher | Matrix theory[END_REF]), we start with the simple case of polynomials.

Definition 1 Let k ∈ N * , and let M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k (their dimensions n 1 , . . . , n k may be different). Let P (x 1 , . . . , x k ) = α∈F c α x α 1 1 . . . x α k k be a polynomial of k variables (here F is a finite subset of N k , c α ∈ C and the α l 's are the coordinates of α). We set:

P ⊗ (M 1 , . . . , M k ) := α∈F c α M α 1 1 ⊗ . . . ⊗ M α k k ∈ k l=1 E l ⊗ E * l .
Remark 2 In this definition, the tensor products are implicitely taken over C. We can use a similar definition if E 1 , . . . , E k are R-vector spaces and if P has real coefficients, but with tensor products over R. In the cases when some of the E l 's are R-vector spaces and the other ones are C-vector spaces, or if all the E l 's are R-vector spaces and P has complex coefficients, we can complexify the real vector spaces by replacing the concerned E l 's by E ′ l = C ⊗ R E l and M l by its natural C-linear extension M ′ l to E ′ l , and use the classical version of definition 1.

Remark 3 We use the notation P ⊗ (M 1 , . . . , M k ) and not P (M 1 , . . . , M k ), because doing so would be confusing, and because the ⊗ sign reminds you it is a tensor of higher order. For example, we have ((x, y) → x + y) ⊗ (A, B) = A ⊗ I + I ⊗ B = A + B in general (even in simple cases where A = B or B = 0). Lemma 4 Let k ∈ N * , and let M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k . For 1 ≤ l ≤ k, let A (l) be an invertible C-linear application from E l to another C-vector space F l of same dimension than E l . We set M ′ l = A (l) M l A -1 (l) ∈ L(F l , F l ). Let P be a polynomial of k variables. Then we have:

P ⊗ (M ′ 1 , . . . , M ′ k ) i 1 j 1 . . . i k j k = A (1) i 1 i ′ 1 A -1
(1)

j ′ 1 j 1 . . . A (k) i k i ′ k A -1 (k) j ′ k j k P ⊗ (M 1 , . . . , M k ) i ′ 1 j ′ 1 . . . i ′ k j ′ k .
Proof: In the simple case where P is a monomial, it follows from the fact that (AM A -1 ) n = AM n A -1 . The more general case where P is any polynomial easyly follows by linearity. Any complex square matrix can be put in a Jordan form by conjugation by an invertible matrix. Let us see what one gets when all the matrices M 1 , . . . , M k are Jordan matrices.

Example 5 Let M 1 , . . . , M k be Jordan matrices, i.e, for 1 ≤ l ≤ k, the matrix M l has the following form:

M l =       J λ l1 ,r l1 0 . . . 0 0 J λ l2 ,r l2 . . . . . . . . . . . . . . . 0 0 . . . 0 J λ lb l ,r lb l      
, where b l is the number of jordan blocks in M l , J λ,r is the r × r square matrix . . , M k ) i 1 j 1 . . . i k j k is given by the following formulas:

J λ,r =          λ 1
• If for some 1 ≤ l ≤ k, one has i l > j l or c l = d l , then the coefficient P ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k is 0.

• If for all 1 ≤ l ≤ k, one has i l ≤ j l and c l = d l , then the coefficient

P ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k is k l=1 1 (j l -i l )! ∂ j l -i l l P (λ 1c 1 , . . . , λ kc k ).
Proof: Like in the proof of Lemma 4, we start with the simple case where P is a monomial. In this case, P (x 1 , . . . ,

x k ) = x α 1 1 . . . x α k k and P ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k = k l=1 M α l l i l j l .
One easyly gets the expression given in Example 5 for the coefficients of P ⊗ (M 1 , . . . , M k ) by using the special form of the coefficients of powers of Jordan matrices, and the fact that

k l=1 ∂ ∂x l a l k l=1 f l (x l ) = k l=1 f (a l ) l (x l
). The case of a more general polynomial P trivially follows by linearity.

Proposition 6 Let k ∈ N * , and let M 1 , . . . , M k be square matrices with complex coefficients. Let P 1 , . . . , P k be their minimal polynomials.

Then the set of polynomials P of k variables such that

P ⊗ (M 1 , . . . , M k ) = 0
is the ideal generated by the polynomials P l (x l ), for 1 ≤ k ≤ l.

Proof:

There exist M ′ 1 , . . . , M ′ k Jordan matrices and A 1 , . . . , A k invertibles such that M ′ l = A l M l A -1 l .
According to Lemma 4, we have

P ⊗ (M 1 , . . . , M k ) = 0 ⇔ P ⊗ (M ′ 1 , . . . , M ′ k ) = 0.
For 1 ≤ l ≤ k, we set λ lm the eigenvalues of M l , and r lm their multiplicity as root of the minimal polynomial P l of M l (here the index m varies from 1 to the number e l of distinct eigenvalues of M l ). Then according to example 5, P ⊗ (M ′ 1 , . . . , M ′ k ) = 0 is equivalent to: for all (m l ) 1≤l≤k , (j l ) 1≤l≤k satisfying 1 ≤ m l ≤ e l and 0 ≤ j l < r lm l , one has

k l=1 ∂ j l l P (λ 1m 1 , . . . , λ km k ) = 0.
Assume that there exists l such that P (x 1 , . . . , x k ) = P l (x l )Q(x 1 , . . . , x k ), where Q is a polynomial. Then for any 1 ≤ m ≤ e l and any 0 ≤ j < m lm , x lλ lm divides ∂ j l P (x 1 , . . . , x k ), so it also divides all the derivatives of ∂ j l P (x 1 , . . . , x k ) with respect to all variables but x l , thus they cancel when x l = λ lm . Thus P (M 1 , . . . , M k ) = 0. By linearity, for any polynomial P in the ideal generated by P 1 (x 1 ), . . . , P k (x k ), we have P (M 1 , . . . , M k ) = 0. Conversely, let P be a polynomial such that P (M 1 , . . . , M k ) = 0. There exists two polynomials Q and R such that P = Q + R, Q belongs to the ideal generated by P 1 (x 1 ), . . . , P k (x k ) and the degree of R with respect to the variable x l is lower than deg(P l ) (in the sense that any monomial

x α 1 1 . . . x α k k in R satisfies α l < deg(P l ))
. This fact can be shown by performing Euclidean division by the Gröbner basis P 1 (x 1 ), . . . , P k (x k ) (this is a Gröbner basis for any monomial order) in the space of polynomials on k variables. For any 1 ≤ l ≤ k, any 1 ≤ m ≤ e l and any 0 ≤ j < r lm , there exists a polynomial of one variable P lmj of degree less than deg(P l ) such that for any 1 ≤ m ′ ≤ e l and any 0 ≤ j ′ < r lm ′ , one has P (j ′ ) lmj (λ lm ′ ) = 1 m=m ′ ,j=j ′ (this follows from the classical theory of Lagrange-Sylvester interpolation polynomials). For any m 1 , . . . , m k and j 1 , . . . , j k such that 1 ≤ m l ≤ e l and 0 ≤ j l < r lm l , we set P m 1 j 1 ...m k j k (x 1 , . . . x k ) = k l=1 P lm l j l (x l ). Let us consider the linear map ν which associates to any polynomial S(x 1 , . . . , x k ) of k variables such that its degree with respect to x l is less than deg(P l ), the following k l=1 deg(P l ) values: for any m 1 , . . . , m k and j 1 , . . . , j k satisfying 1 ≤ m l ≤ e l and 0 ≤ j l < r lm l , we set

ν m 1 j 1 ...m k j k (S) := k l=1 ∂ j l l S(λ 1m 1 , . . . , λ km k ). Then we have ν m 1 j 1 ...m k j k (P m ′ 1 j ′ 1 ...m ′ k j ′ k ) = 1 if m l = m ′ l and j l = j ′ l for every 1 ≤ l ≤ k and ν m 1 j 1 ...m k j k (P m ′ 1 j ′ 1 ...m ′ k j ′
k ) = 0 otherwise, so the linear map ν is surjective. So because of the equality of the (finite) dimensions of its domain and its image, ν is also injective. So since ν(R) = 0, we have R = 0 and thus P = Q belongs to the ideal generated by the P l (x l ).

So one does see from Proposition 6 that the dependancy in P of P (M 1 , . . . , M k ) only relies on the values of P and of some of its derivatives at the eigenvalues of the matrices M l , 1 ≤ l ≤ k. This justifies the following definition for more general functions than polynomials.

Definition 7 Let k ∈ N * , and let M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k . For 1 ≤ l ≤ k, let e l be the number of different eigenvalues of M l , λ lm be the eigenvalues of M l , for 1 ≤ m ≤ e l , and r lm be the multiplicity of the root λ lm in the minimal polynomial of M l . Let f be a function of k complex variables. Assume that for any k-uple of integers (m 1 , . . . , m k ) such that 1 ≤ m l ≤ e l for 1 ≤ l ≤ k, the function f (x 1 , . . . , x k ) is holomorphic with respect to the variables x l with l satisfying r lm l ≥ 2 near (λ 1m 1 , . . . , λ km k ). That is, if we denote by l 1 , . . . , l p the indexes such that r lq ≥ 2 for 1, ≤ q ≤ p, the function

g : C p → C (y 1 , . . . , y p ) → f (λ 1m 1 , . . . , λ km k ) + p q=1 y q v lq
admits a continuous C-linear differential on a neighborhood of 0, where v l is the l-th vector of the canonical basis of C k . Then we set:

f ⊗ (M 1 , . . . , M k ) := P ⊗ (M 1 , . . . , M k ),
with P any complex polynomial of k variables such that for all (m 1 , . . . , m k ) and (j 1 , . . . , j k ) such that 1 ≤ m l ≤ e l and 0 ≤ j l < r lm l for 1 ≤ l ≤ k, we have:

k l=1 ∂ j l l f (λ 1m 1 , . . . , λ km k ) = k l=1 ∂ j l l P (λ 1m 1 , . . . , λ km k ).
Remark 8 The function f does not really need to be defined on the whole C k , but only on the points whose coordinates are the eigenvalues of the M l 's and on some neighborhoods of those points intersected with some affine subspaces.

Remark 9 If one wants to generalize this definition in a real framework, as said in Remark 2, one can not just assume f is a real function of real variables, because real matrices may have non-real eigenvalues. The right thing to do is to chose f such that f ( z1 , . . . , zk ) = f (z 1 , . . . , z k ) to be sure there exists a polynomial P with real coefficients satisfying all the equalities required.

Remark 10 The polynomial P can be chosen to be

k l=1 ∂ j l l f (λ 1m 1 , . . . , λ km k ) P m 1 j 1 ...m k j k (x 1 , . . . , x k ),
with the same notation as in the proof of Proposition 6. In this case, we say P is the Lagrange-Sylvester interpolation polynomial of f associated to the product of multisets k l=1 Root(P l ), where Root(P l ) is the multiset of the roots of P l , counted with their multiplicity.

Some rules for computations

In this section, we show that f ⊗ (M 1 , . . . , M k ) behaves like fonctions of several variables do, and we show that some contractions of f ⊗ (M 1 , . . . , M k ) can have a simplified expression. Finally, we show the main result of this paper, which is the expression of the derivative of f ⊗ (M 1 , . . . , M k ) with respect to the matrices M l .

The tensor f ⊗ (M 1 , . . . , M k ) applied to eigenvectors of M 1 , . . . , M k has a simplified expression. This allows to give an alternative expression for f ⊗ (M 1 , . . . , M k ) when M 1 , . . . , M k are all diagonalizable.

Proposition 11 Let k ∈ N * , and let M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k . For 1 ≤ l ≤ k, let u l ∈ E l be an eigenvector of M l for the eigenvalue λ l . then we have

f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k (u 1 ) j 1 . . . (u k ) j k = f (λ 1 , . . . , λ k )(u 1 ) i 1 . . . (u k ) i k .
Proof: In the simple case where f is a monomial, we have

f (x 1 , . . . , x k ) = x α 1 1 . . . x α k k , with α l ∈ N for 1 ≤ l ≤ k. We get f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k (u 1 ) j 1 . . . (u k ) j k = (M α 1 1 ) i 1 j 1 . . . M α k k i k j k (u 1 ) j 1 . . . (u k ) j k = λ α 1 1 (u 1 ) i 1 . . . λ α k k (u k ) i k = f (λ 1 , . . . , λ k )(u 1 ) i 1 . . . (u k ) i k .
By linearity, this extends to the case where f is a polynomial. For the more general case, we have f ⊗ (M 1 , . . . , M k ) = P ⊗ (M 1 , . . . , M k ) where P is a polynomial of k variables satisfying the conditions described in Definition 7. In particular, P has the same values f has on the eigenvalues of M 1 , . . . , M k , so the desired equality holds for a general function f . Remark 12 In the case where M 1 , . . . , M k are all diagonalizable, for all 1 ≤ l ≤ k we can take a basis of E l made of eigenvectors of M l . Let us denote by u lm , for 1 ≤ m ≤ dim(E l ) = d l the vectors of this basis, by λ lm the corresponding eigenvalue and by u * lm ∈ E * l the corresponding vector of the dual basis.

Then the family of tensors k l=1 u lm l ⊗u * ln l , for 1 ≤ m l ≤ d l and

1 ≤ n l ≤ d l , is a basis of k l=1 E l ⊗E * l .
According to Proposition 11, the decomposition of the tensor f ⊗ (M 1 , . . . , M k ) in this basis can only be

f ⊗ (M 1 , . . . , M k ) = ∀1≤l≤k,1≤m l ≤d l f (λ 1m 1 , . . . , λ km k ) k l=1 u lm l ⊗ u * lm l .
One of the simplest properties of f ⊗ (M 1 , . . . , M k ) is its linearity with respect to f . Proposition 13 Let k ∈ N * , and let M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k . Let λ and µ be two complex numbers, and f and g be two functions from C k to C, regular enough such that f ⊗ (M 1 , . . . , M k ) and g ⊗ (M 1 , . . . , M k ) can be defined. Then the function λf + µg has the same regularity, and we have:

(λf + µg) ⊗ (M 1 , . . . , M k ) = λf ⊗ (M 1 , . . . , M k ) + µg ⊗ (M 1 , . . . , M k ).
Proof: If P and Q are interpolation polynomials of f and g as required in Definition 7 to compute f ⊗ (M 1 , . . . , M k ) and g ⊗ (M 1 , . . . , M k ), then λP +µQ is an interpolation polynomial of λf + µg because partial derivatives are linear.

So we have

(λf + µg) ⊗ (M 1 , . . . , M k ) = (λP + µQ) ⊗ (M 1 , . . . , M k ) = λP ⊗ (M 1 , . . . , M k ) + µQ ⊗ (M 1 , . . . , M k ) = λf ⊗ (M 1 , . . . , M k ) + µg ⊗ (M 1 , . . . , M k ),
where the second equality trivially follows from Definition 1. Another trivial property is the nice behaviour of f ⊗ (M 1 , . . . , M k ) with respect to transpositions.

Proposition 14 Let k ∈ N * , and let M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k . Let f be a function from

C k to C such that f ⊗ (M 1 , . . . , M k ) is well defined. Let 1 ≤ l ≤ k. The transposition of M l is the endomorphism of E * l defined by M T l (ζ), v = ζ, M l (v) , for any ζ in E * l and v ∈ E l . More explicitely, we can write (M T l ) j l i l = (M l ) i l j l .
Then we have:

f ⊗ (M 1 , . . . , M l-1 , M T l , M l+1 , . . . , M k ) i 1 j 1 . . . i l-1 j l-1 j l i l i l+1 j l+1 . . . i k j k = f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k .
Remark 15 In particular, if the matrix of M l is symmetric in some basis B of E l , then the coefficients of f ⊗ (M 1 , . . . , M k ) in a product basis where the one chosen for E l is B are invariant by swapping the corresponding indexes.

Proof: The matrices M l and M T l have the same eigenvalues and the same minimal polynomial, thus if P is a suitable interpolation polynomial so that

f ⊗ (M 1 , . . . , M k ) = P ⊗ (M 1 , . . . , M k ), then we have f ⊗ (M 1 , . . . , M l-1 , M T l , M l+1 , . . . , M k ) = P ⊗ (M 1 , . . . , M l-1 , M T l , M l+1 , . . . , M k ).
Proposition 14 for polynomials trivially follows from the fact that (M T ) a = (M a ) T .

The tensor f ⊗ (M 1 , . . . , M k ) can also be seen as an endomorphism on the space E 1 ⊗ E 2 ⊗ . . . ⊗ E k . This allows to take products of such tensors, or to apply functions of several variables on them.

Theorem 16 Let M 1 , . . . , M k be endomorphisms of finite dimensional Cvector spaces. Let f 1 and f 2 be two functions from C k to C. As in Definition 7, we set e l the number of different eigenvalues of M l , for 1 ≤ l ≤ k, λ lm the eigenvalues of M l and r lm their multiplicity as roots of the minimal polynomial of M l , for 1 ≤ m ≤ e l . Assume that f 1 and f 2 are holomorphic with respect to all the variables x l such that r lm l ≥ 2 near (λ 1m 1 , . . . , λ km k ).

Then we can define

M1 = f ⊗ 1 (M 1 , . . . , M k ) and M2 = f ⊗ 2 (M 1 , . . . , M k ) and see them as endomorphisms of E 1 ⊗ . . . ⊗ E k . Then we have M1 M2 = g ⊗ (M 1 , . . . , M k ), with g(x 1 , . . . , x k ) = f 1 (x 1 , . . . , x k )f 2 (x 1 , . . . , x k ).
Remark 17 In particular, M1 and M2 commute, since one does get the same function g by swapping f 1 and f 2 .

Proof:

We first notice that the function g is regular enough so that g ⊗ (M 1 , . . . , M k ) is well defined, so the statement of Proposition 16 has a sense.

Let P 1 and P 2 be suitable interpolation polynomials of f 1 and f 2 , in the sense that all the partial derivatives at (λ 1m 1 , . . . , λ km k ) such that one derives less than r lm l times with respect to x l are equal for f i and P i .

Then according to Definition 7, one has

f ⊗ i (M 1 , . . . , M k ) = P ⊗ i (M 1 , . . . , M k
). The polynomials P 1 and P 2 can be written as

P i = α∈F i a iα x α 1 1 . . . x α k k ,
with F i a finite subset of N k and a iα the complex coefficients of the polynomial P i . Then we have:

( M1 M2 ) i 1 j 1 . . . i k j k = ( M1 ) i 1 m 1 . . . i k m k ( M2 ) m 1 j 1 . . . m k j k = α∈F 1 a 1α (M α 1 1 ) i 1 m 1 . . . (M α k k ) i k m k β∈F 2 a 2β (M β 1 1 ) m 1 j 1 . . . (M β k k ) m k j k = α∈F 1 ,β∈F 2 a 1α a 2β (M α 1 +β 1 1 ) i 1 j 1 . . . (M α k +β k k ) i k j k = (P 1 P 2 ) ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i k j k .
Finally, we have

∂ a 1 1 , . . . , ∂ a k k (P 1 P 2 )(x 1 , . . . , x k ) = 0≤b 1 ≤a 1 ... 0≤b k ≤a k a 1 b 1 . . . a k b k ∂ b 1 1 . . . ∂ b k k P 1 (x 1 , . . . , x k )∂ a 1 -b 1 1 . . . ∂ a k -b k k P 2 (x 1 , . . . , x k ).
the same formula holds when we replace P 1 with f 1 , P 2 with f 2 , P 1 P 2 with g, and x l with λ lm l , for all k-tuple (m 1 , . . . , m k ) such that 1 ≤ m l ≤ e l , provided 0 ≤ a l < r lm l for all 1 ≤ l ≤ k. Thus we have

∂ a 1 1 . . . ∂ a k k g(λ 1m 1 , . . . , λ km k ) = ∂ a 1 1 . . . ∂ a k k (P 1 P 2 )(λ 1m 1 , . . . , λ km k ),
for all k-tuple (a 1 , . . . , a k ) such that 0 ≤ a l < r lm l . So we get

M1 M2 = (P 1 P 2 ) ⊗ (M 1 , . . . , M k ) = g ⊗ (M 1 , . . . , M k ),
as stated.

Theorem 18 Let r ∈ N * , and k q ∈ N * for 1 ≤ q ≤ r. Let E ql be a family of finite dimensional C-vector spaces, where 1 ≤ l ≤ r and 1 ≤ l ≤ k q , and let M ql be and endomorphism of E ql .

We set e ql the number of different eigenvalues of M ql , λ qlm these eigenvalues, where 1 ≤ m ≤ e ql , and r qlm the multiplicity of the root λ qlm in the minimal polynomial of M ql . For 1 ≤ q ≤ r, let f q be a function from C kq to C, such that f q is holomorphic with respect to all the variables x l such that r qlm l ≥ 2 near (λ q1m 1 , . . . , λ qlm l , . . . , λ qkqm kq ), for every k q -tuple (m 1 , . . . , m kq ) with 1 ≤ m l ≤ e ql .

Let ēq be the number of different values that f q (λ q1m 1 , . . . , λ qkqm kq ) takes, and µ qp be these values for 1 ≤ p ≤ ēq .

We set rqp := max{1+ kq l=1 (r qlm l -1), 1 ≤ m l ≤ e ql |f q (λ q1m 1 , . . . , λ qkqm kq ) = µ qp }.

We set Mq = f ⊗ q (M q1 , . . . , M qkq ), seen as an endomorphism of Ēq := E q1 ⊗ . . . ⊗ E qkq .

Let g be a function from C r to C, such that for every r-tuple (p 1 , . . . , p r ), g is holomorphic with respect to all the variables x q such that rqpq ≥ 2, near the point (µ 1p 1 , . . . , µ rpr ).

Then for all 1 ≤ q ≤ k, and all 1 ≤ p ≤ ēq , the µ qp 's are the eigenvalues of Mq , and the multiplicity of the root µ qp in the minimal polynomial of Mq is at most rq p.

So g ⊗ ( M1 , . . . , Mr ) is well defined and furthermore, we have:

g ⊗ ( M1 , . . . , Mr ) = h(M 11 , . . . , M rkr ),
where h is the function of r q=1 k q variables defined by h(x 11 , . . . , x rkr ) := g(f 1 (x 11 , . . . , x 1k 1 ), . . . , f r (x r1 , . . . , x rkr )).

Proof: Let us prove that the µ qp 's are the eigenvalues of Mq and have multiplicity at most rpq in the minimal polynomial of Mq . Let 1 ≤ q ≤ r. Let P (x 1 , . . . , x kq ) be a suitable interpolation polynomial of f q , such that Mq = P ⊗ (M q1 , . . . , M qkq ). For any m 1 , . . . , m kq , if v 1 , . . . , v kq are eigenvectors of M q1 , . . . , M qkq for the eigenvalues λ q1m 1 , . . . , λ qkqm kq , then v 1 ⊗. . .⊗v kq is an eigenvector of Mq for the eigenvalue P (λ q1m 1 , . . . , λ qkqm kq ) = f q (λ q1m 1 , . . . , λ qkqm kq ), so the µ qp 's are eigenvalues of Mq . Let Q(x) = ēq p=1 (xµ qp ) rqp . It remains to check that Q( Mq ) = 0. We set R(x 1 , . . . , x kq ) = Q(P (x 1 , . . . , x kq )). Then it easily follows from Theorem 16 that Q( Mq ) = R ⊗ (M q1 , . . . , M qkq ). For 1 ≤ l ≤ k q , let 1 ≤ m l ≤ e ql , and 0 ≤ j l < r qlm l . One has, according to the Faa di Bruno formula,

  kq l=1 ∂ j l l   R(λ q1m 1 , . . . , λ qkqm kq ) = kq l=1 j l j=0 Q (j) (P (λ q1m 1 , . . . , λ qkqm kq ))A j ,
where the term A j can be written as

A j = n≤j (i 1 ,...,in)∈(N kq ) n 0≺i 1 ≺i 2 ≺...≺in (a 1 ,...,an)∈N * n a 1 +...+an=j a 1 i 1 +...+anin=(j 1 ,...,j kq ) kq l=1 j l ! n k=1 a k ! kq l=1 (i kl !) a k n k=1     kq l=1 ∂ i kl l   P (λ q1m 1 , . . . , λ qkqm kq )   a k ,
where can be any total order on N kq such that 0 is its minimal element (one can take the lexicographical order, for example).

But actually the value of A j does not matter, since Q (j) (P (λ q1m 1 , . . . , λ qkqm kq )) = 0, because P (λ q1m 1 , . . . , λ qkqm kq ) is equal to some µ qp such that rqp > j. for all p 1 , . . . , p r and j 1 , . . . , j r satisfying 1 ≤ p q ≤ ēq and 0 ≤ j q < rqpq .

Thus, we have

Then we have P ⊗ ( M1 , . . . , Mr ) = g ⊗ ( M1 , . . . , Mr ). It easily follows from Definition 1 and Theorem 16 that

P ⊗ ( M1 , . . . , Mr ) = H ⊗ (M 11 , . . . , M rkr ),
where H is the function of r q=1 k q variables defined by H(x 11 , . . . , x rkr ) = P (f 1 (x 11 , . . . , x 1k 1 ), . . . , f r (x r1 , . . . , x rkr )).

To complete the proof of the theorem, it remains to check that for all 1 ≤ m ql ≤ e ql and 0 ≤ j ql < r qlm ql . This is easily done by using the Faa di Bruno formula and the fact that

  r q=1 ∂ jq q   P (µ 1p 1 , . . . , µ rpr ) =   r q=1 ∂ jq q   g(µ 1p 1 , . . . , µ rpr ),
for all 1 ≤ p q ≤ ēq and 0 ≤ j q < rqpq .

In some cases, contractions of the tensor f ⊗ (M 1 , . . . , M k ) have a simplified expression.

Theorem 19 Let M 1 , . . . , M k be endomorphisms of finite dimensional Cvector spaces. Let f be a function from

C k to C, such that f ⊗ (M 1 , . . . , M k ) is well defined. Let 1 ≤ p ≤ k.
Then we have the following equality:

f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 ip ip i p+1 j p+1 . . . i k j k = g ⊗ (M 1 , . . . , M p-1 , M p+1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i p+1 j p+1 . . . i k j k ,
where g is the function of k -1 variables defined by

g(x 1 , . . . , x p-1 , x p+1 , . . . , x k ) = ep m=1 s pm f (x 1 , . . . , x p-1 , λ pm , x p+1 , . . . , x k ),
where the λ pm , for 1 ≤ m ≤ e p are all the different eigenvalues of M p and s pm is the multiplicity of the eigenvalue λ pm , i.e. its multiplicity as root of the characteristic polynomial of M p .

Proof: In the simple case where f (x 1 , . . . , x k ) is a monomial, the equality of Theorem 19 easily follows from the classical fact that Tr(M a ) = λ eigenvalue of M λ a (where the same λ appears several times in the sum if it is a multiple eigenvalue of M ) for any square matrice M . By linearity, the equality of Theorem 19 extends to the case where f is a polynomial.

For a more general f , we set P a polynomial such that k l=1 ∂ a l l (f -P )(λ 1m 1 , . . . , λ km k ) = 0 for any sequences m 1 , . . . , m k and a 1 , . . . , a k such that 1 ≤ m l ≤ e l and 0 ≤ a l < r lm l , with e l the number of different eigenvalues of M l , λ lm these eigenvalues for 1 ≤ m ≤ e l and r lm their multiplicities in the minimal polynomial of M l . Then we have

f ⊗ (M 1 , . . . , M k ) = P ⊗ (M 1 , . . . , M k ). Thus we get f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 ip ip i p+1 j p+1 . . . i k j k = P ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 ip ip i p+1 j p+1 . . . i k j k = Q ⊗ (M 1 , . . . , M p-1 , M p+1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i p+1 j p+1 . . . i k j k , with Q(x 1 , . . . , x p-1 , x p+1 , . . . , x k ) = ep m=1
s pm P (x 1 , . . . , x p-1 , λ pm , x p+1 , . . . , x k ).

To conclude the proof, it remains to check that

    1≤l≤k l =p ∂ ∂x l a l     (g -Q)(x 1 , . . . , x p-1 , x p+1 , . . . , x k ) (x 1 ,...,x k )=(λ 1m 1 ,...,λ km k ) = 0.
This quantity is just

ep m=1     1≤l≤k l =p ∂ ∂x l a l     (f -P )(λ 1m 1 , . . . , λ p-1m p-1 , λ pm , λ p+1m p+1 , . . . , λ km k ), which is trivially 0. So Q ⊗ (M 1 , . . . , M p-1 , M p+1 , . . . , M k ) = g ⊗ (M 1 , . . . , M p-1 , M p+1 , .
. . , M k ) and the theorem is proved.

Theorem 20 Let M 1 , . . . , M k be endomorphisms of finite dimensional Cvector spaces E 1 , . . . , E k . Assume that for two indexes 1 ≤ p < q ≤ k, we have E p = E q and M p = M q . Let f be a function from C k to C, such that f ⊗ (M 1 , . . . , M k ) is well defined. Then we have:

f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i a i p+1 j p+1 . . . i q-1 j q-1 a j i q+1 j q+1 . . . i k j k = f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1
a j i p+1 j p+1 . . . i q-1 j q-1 i a i q+1 j q+1 . . . i k j k = g ⊗ (M 1 , . . . , M q-1 , M q+1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i j i p+1 j p+1 . . . i q-1 j q-1 i q+1 j q+1 . . .

i k j k ,
where the function g of k -1 variables is defined by g(x 1 , . . . , x q-1 , x q+1 , . . . , x k ) = f (x 1 , . . . , x q-1 , x p , x q+1 , . . . , x k ).

Proof: We proceed as in the proof of Theorem 19. If f is a monomial, the result easily follows from the trivial fact that M a M b = M b M a = M a+b for any square matrix M . So by linearity, it also holds for polynomials. For a more general f , we set again P an interpolation polynomial such that f ⊗ (M 1 , . . . , M k ) = P ⊗ (M 1 , . . . , M k ). And so we have:

f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i a i p+1 j p+1 . . . i q-1 j q-1 a j i q+1 j q+1 . . . i k j k = P ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i a i p+1 j p+1 . . . i q-1 j q-1 a j i q+1 j q+1 . . . i k j k = P ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 a j i p+1 j p+1 . . . i q-1 j q-1 i a i q+1 j q+1 . . . i k j k = f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 a j i p+1 j p+1 . . . i q-1 j q-1 i a i q+1 j q+1 . . . i k j k = Q ⊗ (M 1 , . . . , M q-1 , M q+1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i j i p+1 j p+1 . . . i q-1 j q-1 i q+1 j q+1 . . . i k j k ,
where the polynomial Q is defined by Q(x 1 , . . . , x q-1 , x q+1 , . . . , x k ) = P (x 1 , . . . , x q-1 , x p , x q+1 , . . . , x k ).

It remains to check that

    1≤l≤k l =q ∂ ∂x l a l     (g -Q)(x 1 , . . . , x q-1 , x q+1 , . . . , x k ) (x 1 ,...,x k )=(λ 1m 1 ,...,λ km k ) = 0,
with the same notation as above for the λ lm 's and the same conditions for the m l 's and a l 's. The left-hand side is equal to

ap h=0 a p h       ∂ h p ∂ ap-h q 1≤l≤k l =p l =q ∂ a l l       (f -P )(λ 1m 1 , . . . , λ q-1mq 1 , λ pmp , λ q+1m q+1 , . . . , λ km k ).
It is 0 because P is a nice interpolation polynomial of f . So we get Q ⊗ (M 1 , . . . , M q-1 , M q+1 , . . . , M k ) = g ⊗ (M 1 , . . . , M q-1 , M q+1 , . . . , M k ) and the theorem is proved.

In the case where E p = E q but M p and M q are different, there is no such simplified expression for the contraction, but one has a kind of commutation property if M p and M q commute. Proposition 21 Let M 1 , . . . , M k be endomorphisms of finite dimensional C-vector spaces E 1 , . . . , E k . Assume that for two indexes 1 ≤ p < q ≤ k, we have E p = E q and the endomorphisms M p and M q commute. Let f be a function from C k to C, such that f ⊗ (M 1 , . . . , M k ) is well defined. Then we have:

f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 i a i p+1 j p+1 . . . i q-1 j q-1 a j i q+1 j q+1 . . . i k j k = f ⊗ (M 1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 a j i p+1 j p+1 . . . i q-1 j q-1 i a i q+1 j q+1 . . . i k j k .
Proof: One replaces f with a suitable interpolation polynomial P such that f ⊗ (M 1 , . . . , M k ) = P ⊗ (M 1 , . . . , M k ). Then using linearity, we only have to check the property for monomials, which easily follows from the well known fact that if M p and M q commute, then M n 1 p and M n 2 q also commute (the commutation of two endomorphisms M and N can be written M i a N a j = M a j N i a ).

Remark 22

We have a generalization of Remark 17. If M 1 , . . . , M k and M ′ 1 . . . , M ′ k are endomorphisms of E 1 , . . . , E k such that for all 1 ≤ l ≤ k, M l and M ′ l commute, and if f and g are two functions from C k to C such that M = f ⊗ (M 1 , . . . , M k ) and M ′ = g ⊗ (M ′ 1 , . . . , M ′ k ) are well defined, then M and M ′ commute as endomorphisms of E 1 ⊗ . . . ⊗ E k .

Proof:

The tensors M M ′ and M ′ M can both be obtained by k contractions from the tensor h ⊗ (M 1 , . . . , M k , M ′ 1 , . . . , M ′ k ), where h(x 1 , . . . , x k , y 1 , . . . , y k ) = f (x 1 , . . . , x k )g(y 1 , . . . , y k ). The fact you get the same result follows just from applying k times proposition 21.

For a given holomorphic function f , the tensor f ⊗ (M 1 , . . . , M k ) is an holomorphic function of M 1 , . . . , M k . The following theorem gives the expression of its derivatives.

Theorem 23 Let k ∈ N * , and M 1 , . . . , M k be endomorphisms of the finite dimensional C-vector spaces E 1 , . . . , E k . For any 1 ≤ l ≤ k, we set e l the number of different eigenvalues of M l , λ lm these eigenvalues for 1 ≤ m ≤ e l , r lm their multiplicity as roots of the minimal polynomial of M l . Let f be a function from C k to C. Let 1 ≤ p ≤ k. Assume that for all k-tuple (m 1 , . . . , m k ), f (x 1 , . . . , x k ) is holomorphic with respect to x p and all the other variables x l such that r lm l ≥ 2 near (λ 1m 1 , . . . , λ km k ).

Then for any endomorphism H of E p , we have the following:

lim ε→0 f ⊗ (M 1 ,...,M p-1 ,Mp+εH,M p+1 ,...,M k ) i 1 j 1 ... i k j k -f ⊗ (M 1 ,...,...,M k ) i 1 j 1 ... i k j k ε = g ⊗ (M 1 , . . . , M p-1 , M p , M p , M p+1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 ip i j jp i p+1 j p+1 . . . i k j k H i j ,
with g the function of k + 1 variables defined by

g(x 1 , . . . , x p-1 , x p , y, x p+1 , . . . , x k ) = f (x 1 ,...,x p-1 ,y,x p+1 ,...,x k )-f (x 1 ,...,x k ) y-xp if y = x p ∂ p f (x 1 , . . . , x k ) if y = x p .
Remark 24 When y is close to x, we can write

g(x 1 , . . . , x p-1 , x p , y, x p+1 , . . . , x k ) = 1 0 ∂ p f (x 1 , . . . , x p-1 , (1-t)x p +ty, x p+1 , . . . , x k ).
So we don't have regularity issues with g near the hyperplane x p = y.

Proof: In the simple case of a monomial f (x 1 , . . . , x k ) = k l=1 x α l l , it follows from the fact that lim ε→0

((M p + εH) αp ) i j -(M αp p ) i j ε = αp-1 h=0 (M h p HM αp-1-h p ) i j =   (x, y) → αp-1 h=0 x h y αp-1-h   ⊗ (M p , M p ) i a b j H a b
and that we have

αp-1 h=0 x h y αp-1-h = y αp -x αp y-x if x = y d dx x αp if x = y.
By linearity, this result extends to polynomials. For a more general f , for any 1 ≤ l ≤ k, we set P l the minimal polynomial of M l , and we set P (ε) the characteristic polynomial of M p + εH. Let Q ε be the Lagrange-Sylvester interpolation polynomial of f associated to the product of multisets p-1 l=1 Root(P l )×Root(P p P (ε) )× k l=p+1 Root(P l ). That is to say, if we set e l the number of different roots of P l , λ lm these roots for 1 ≤ m ≤ e l and r lm their multiplicities, and if we set likewise e (ε) the number of different roots of P p P (ε) , λ (ε) m these roots for 1 ≤ m ≤ e (ε) and r (ε) m their multiplicities, then Q ε (x 1 , . . . , x k ) is the unique polynomial of k variables whose degree with respect to the variable x l is at most deg(P l ), except for the variable x p for which it is at most deg(P p ) + dim(E p ), and wich satisfies the equalities

k l=1 ∂ a l l (Q ε -f )(λ 1m 1 , . . . , λ p-1m p-1 , λ (ε)
mp , λ p+1m p+1 , . . . , λ km k ) = 0 for all k-tuples (m 1 , . . . , m k ) and (a 1 , . . . , a k ) such that 1 ≤ m l ≤ e l and 0 ≤ a l < r lm l for l = p and 1 ≤ m p ≤ e (ε) and 0 ≤ a p < r (ε) mp . Because of the continuity of the Lagrange-Sylvester interpolation, we have

Q ε ---→ ε→0 Q 0 . Furthermore, we have f ⊗ (M 1 , . . . , M p-1 , M p +εH, M p+1 , . . . , M k ) = Q ⊗ ε (M 1 , . . . , M p-1 , M p +εH, M p+1 , . . . , M k ). So we get f ⊗ (M 1 , . . . , M p-1 , M p + εH, M p+1 , . . . , M k ) -f ⊗ (M 1 , . . . , M k ) = Q ⊗ ε (M 1 , . . . , M p-1 , M p + εH, M p+1 , . . . , M k ) -Q ⊗ 0 (M 1 , . . . , M k ) = (Q ε -Q 0 ) ⊗ (M 1 , . . . , M p-1 , M p + εH, M p+1 , . . . , M k ) + Q ⊗ 0 (M 1 , . . . , M p-1 , M p + εH, M p+1 , . . . , M k ) -Q ⊗ 0 (M 1 , . . . , M k ).
For a fixed ε, the difference

(Q ε -Q 0 ) ⊗ (M 1 , . . . , M p-1 , M p +δH, M p+1 , . . . , M k )
is a polynomial of δ whose coefficients tend to 0 when ε tends to 0. The coefficient of this polynomial corresponding to

δ 0 is (Q ε -Q 0 ) ⊗ (M 1 , . . . , M k ) = 0. Thus we have (Q ε -Q 0 ) ⊗ (M 1 , . . . , M p-1 , M p + εH, M p+1 , . . . , M k ) = o(ε).
Hence, using the Theorem for the polynomial Q 0 , we have

lim ε→0 f ⊗ (M 1 ,...,M p-1 ,Mp+εH,M p+1 ,...,M k ) i 1 j 1 ... i k j k -f ⊗ (M 1 ,...,...,M k ) i 1 j 1 ... i k j k ε = R ⊗ (M 1 , . . . , M p-1 , M p , M p , M p+1 , . . . , M k ) i 1 j 1 . . . i p-1 j p-1 ip i j jp i p+1 j p+1 . . . i k j k H i j ,
with R the polynomial of k + 1 variables given by: R(x 1 , . . . , x p-1 , x p , y, x p+1 , . . . , x k ) =

Q 0 (x 1 ,...,x p-1 ,y,x p+1 ,...,x k )-Q 0 (x 1 ,...,x k ) y-xp if y = x p ∂ p Q 0 (x 1 , . . . , x k ) if y = x p .
Let (m 1 , . . . , m p , m ′ p , m p+1 , . . . , m k ) and (a 1 , . . . , a p , a ′ p , a p+1 , . . . , a k ) be two k + 1-tuples satisfying 1 ≤ m l ≤ e l , 1 ≤ m ′ p ≤ e p , 0 ≤ a l < r lm l and 0 ≤ a ′ p < r pm ′ p . We want to show that

∂ ∂y a ′ p k l=1 ∂ ∂x l a l (g -R)(x 1 , . . . , x p , y, x p+1 , . . . , x k ) = 0. (x 1 ,...,xp,y,x p+1 ,...,x k )=(λ 1m 1 ,...,λpm p ,λ pm ′ p ,λ p+1m p+1 ,...,λ km k ) If m p = m ′ p , this quantity is a ′ p h=0 ( a ′ p h ) (-1) a ′ p -h (ap+a ′ p -h)!     ∂ h p 1≤l≤k l =p ∂ a l l     (f -Q 0 )(λ 1m 1 ,...,λ p-1m p-1 ,λ pm ′ p ,λ p+1m p+1 ,...,λ km k ) (λ pm ′ p -λpm p ) ap+a ′ p -h+1 - ap h=0 ( ap h ) (-1) a ′ p (ap+a ′ p -h)!     ∂ h p 1≤l≤k l =p ∂ a l l     (f -Q 0 )(λ 1m 1 ,...,λ km k ) (λ pm ′ p -λpm p ) ap+a ′ p -h+1
which is 0 as wanted, since each term of both sums is 0. If m p = m ′ p , using the formula of Remark 24, the quantity we want to compute is

    ∂ ap+a ′ p +1 p 1≤l≤k l =p ∂ a l l     (f -Q 0 )(λ 1m 1 , . . . , λ km k ) 1 0 (1 -t) a ′ p t ap dt
which is 0 too because a p + a ′ p + 1 < 2r pmp ≤ r pmp + s pmp , where s pmp is the multiplicity of λ pmp as root of the characteristic polynomial of M p .

Thus (g -R) ⊗ (M 1 , . . . , M p-1 , M p , M p , M p+1 , . . . , M k ) = 0 and the theorem is proved.

Some possible applications

In the particular case of symmetric square matrices with real coefficients, we know that such matrices have real eigenvalues and are diagonalizable in an othonormal basis. So we can apply real-valued functions of real variables to them, without having the regularity concerns for the definitions. In this framework, we have the following result.

Proposition 25 We denote by Sym n (R) the vector space of symmetric n×n matrices with real coefficients, equipped with the Hilbert-Schmidt norm

M HS = Tr(M T M ) = Tr(M 2 ).
Let f : R → R be a function. Assume that f is k-Lipschitz for some k > 0.

Then the function

F : Sym n (R) → Sym n (R) M → f (M )
is also k-Lipschitz.

Proof: We first prove that the result holds when f is a C 1 function. In that case, a modified version of Theorem 23 holds, so F is differentiable, and its derivative at M is given by

(dF (M ).H) i j = lim ε→0 F (M + εH) i j -F (M ) i j ε = f 1 (M, M ) i k l j H k l ,
where we have

f 1 (x, y) = f (y)-f (x) y-x if x = y f ′ (x) if x = y.
For a given M ∈ Sym n (R), we denote λ 1 , λ 2 , . . . , λ n its eigenvalues and u 1 , . . . , u n a set of orthonormed eigenvectors of M . Then, the matrices u i ⊗u * i for 1 ≤ i ≤ n and

u i ⊗u * j +u j ⊗u * i √ 2
for 1 ≤ i < j ≤ n form an orthonormal basis of Sym n (R) and are eigenvectors of dF (M ) with eigenvalues f 1 (λ i , λ i ) and f 1 (λ i , λ j ) respectively, according to Proposition 11. Using the fact that

f is k-Lipschitz, we have ∀x, y ∈ R, |f 1 (x, y)| ≤ k.
Thus dF (M ) is a k-Lipschitz linear function of Sym n (R). This being true for every M ∈ Sym n (R), the function F itself is also k-Lipschitz.

If the function f is not C 1 , one can approximate f by a C 1 which is also k-Lipschitz. For example, we set

g(x) =      0 if |x| ≥ 1 e 1 1-x 2 1 -1 e 1 1-y 2 dy if |x| < 1, g ε (x) = 1 ε g x ε and f ε (x) = R f (x -y)g ε (y)dy = R f (y)g ε (x -y)dy.
Then we have

|f ε (x)-f ε (y)| = | R (f (x-z)-f (y-z))g ε (z)dz| ≤ k|x-y| R |g ε (z)|dz = k|x-y|, so f ε is k-Lipschitz. And f ε is differentiable, its derivative being f ′ ε (x) = R f (y)g ′ ε (x -y)dy = R f (x y)g ′ ε (y)dy, which is continuous so f ε is C 1 . Furthermore, we have |f ε (x)-f (x)| = | R (f (y)-f (x))g ε (x-y)dy| ≤ k R |y|g ε (y)dy = kε R |y|g(y)dy.
So for any M ∈ Sym n (R), we have

f ε (M )-f (M ) 2 HS = (f ε -f )(M ) 2 HS = Tr((f ε -f ) 2 (M )) = n i=1 (f ε -f ) 2 (λ i ) ≤ k 2 ε 2 n R |x|g(x)dx 2 .
So, for M 1 and M 2 in Sym n (R), we have

F (M 1 )-F (M 2 ) HS ≤ F (M 1 )-f ε (M 1 ) HS + f ε (M 1 )-f ε (M 2 ) HS + f ε (M 2 )-F (M 2 ) HS ≤ kε √ n R |x|g(x)dx + k M 2 -M 1 HS + kε √ n R |x|g(x)dx.
This inequality being true for every ε > 0, we finally get

F (M 1 )-F (M 2 ) HS ≤ k M 2 -M 1 HS , so F is k-Lipschitz.
The main reason the author thinks it is a good idea to introduce f ⊗ (M 1 , . . . , M k ) is that it allows to give a rather simple expression of derivatives of a function of a square matrix. If we iterate Theorem 23, we can get the following expression for the n-th derivative of f (M ) = f ⊗ (M ).

Proposition 26 Let M and H be two endomorphisms of a finite-dimensional C-vector space E. Let f : C → C be a function which is holomorphic in the neighborhood of each eigenvalue of M . Then the function F : U ⊂ C → L(E, E) given by F (z) = f (M + zH) is holomorphic in a neighborhood of 0 and furthermore, we have:

F (n) (z) i j = n!f ⊗ n (M + zH, M + zH, . . . , M + zH n+1 times ) i j 1 i 1 j 2 . . . i n-1 jn in j H j 1 i 1 . . . H jn in ,
where U is an open subset of C which contains 0, and

f n (x 0 , . . . , x n ) = f [x 0 , . . . , x n ]
is the (generalized) divided difference of the function f on the nodes x 0 , . . . , x n .

Proof: We proceed by induction on n. The case n = 0 is trivial. Assume that

F (n) (z) i j = n!f ⊗ n (M + zH, . . . , M + zH) i j 1 i 1 j 2 . . . i n-1 jn in j H j 1 i 1 . . . H jn in .
then if we want to differentiate this expression one more time with respect to z, we have to differentiate f ⊗ n (M + zH, . . . , M + zH) with respect to each (matricial) variable. So using Theorem 23, we get

F (n+1) (z) i j = n! n k=0 f ⊗ n+1 (M + zH, . . . , M + zH) i j 1 i 1 j 2 . . . i k i ′ j ′ j k+1 . . . i n-1 jn in j H i ′ j ′ H j 1 i 1 . . . H jn in = n! n k=0 f ⊗ n+1 (M + zH, . . . , M + zH) i j 1 i 1 j 2 . . . in j n+1 i n+1 j H j 1 i 1 . . . H j n+1 i n+1 = (n + 1)!f ⊗ n+1 (M + zH, . . . , M + zH) i j 1 i 1 j 2 . . . in j n+1 i n+1 j H j 1 i 1 . . . H j n+1 i n+1 ,
where we just relabelled the indexes to derive line 2 from line 1. So the induction hypothesis is true at the rank n + 1.

Remark 27 In the case where H and M commute, we can get a simpler expression. Indeed, we can write

F (n) (z) i j = n!g ⊗ n (M +zH, . . . , M +zH, H, . . . , H) i j 1 i 1 j 2 . . . i n-1 jn in j j 1 i 1 . . . jn in ,
where g n is the function of 2n + 1 variables defined by g n (x 0 , . . . , x n , y 1 , . . . , y n ) = f [x 0 , . . . , x n ]y 1 . . . y n .

Since H and M commute, H and M + zH commute too. Using Proposition 21 sufficiently many times, we can get

F (n) (z) i j = n!g ⊗ n (M + zH, . . . , M + zH, H, . . . , H) i j 1 j 1 i 1 i 1 j 2 . . . jn in in j .
Using Theorem 20, one gets:

F (n) (z) i j = n!ḡ ⊗ n (M + zH, H) i k k j
where ḡn (x, y) = g n (x, . . . , x, y, . . . , y) = f [x, . . . , x]y n = f (n) (x) n! y n . So finally, we get:

F (n) (z) = f (n) (M + zH)H n .
Remark 28 If we differentiate Tr(f (M + zH)), one gets

F (n) (z) i i = n!f ⊗ n (M + zH, . . . , M + zH) i j 1 i 1 j 2 . . . i n-1 jn in i H j 1 i 1 . . . H jn in .
Using Theorem 20, we get

F (n) (z) i i = n!h ⊗ n,n (M + zH, . . . , M + zH) in j 1 i 1 j 2 . . . i n-1 jn H j 1 i 1 . . . H jn in ,
with h n,n the function of n variables defined by h n,n (x 1 , . . . , x n ) = f n (x n , x 1 , . . . , x n ).

With n = 1, one gets the classical result that

d dz Tr(f (M + zH)) = Tr(f ′ (M + zH)H).
If one differentiates n -1 extra times this formula, one gets

F (n) (z) i i = (n-1)!f ′⊗ n-1 (M +zH, . . . , M +zH) in j 1 i 1 j 2 . . . i n-1 jn H j 1 i 1 . . . H jn in ,
which seems to be different from the other formula above. In fact we get the same thing because if we take h n,k (x 1 , . . . , x n ) = h n,n (x k+1 , . . . , x n , x 1 , . . . , x k ), we have

h ⊗ n,n (M + zH, . . . , M + zH) in j 1 i 1 j 2 . . . i n-1 jn H j 1 i 1 . . . H jn in = h ⊗ n,k (M + zH, . . . , M + zH) in j 1 i 1 j 2 . . . i n-1 jn H j 1 i 1 .
. . H jn in by relabelling the indexes, and the equality

f ′ n-1 (x 1 , . . . , x n ) = n k=1 h n,k (x 1 , . . . , x n ) holds.
Remark 29 Despite the fact that the tensor f ⊗ n (M, . . . , M ) has a lot of symmetries (due to the fact that f n is a symmetric function), the n-linear application F (n) (0).(H 1 , . . . , H n ) obtained by polarization is not in general given by n! times this tensor, but by a symmetrization of it.

F (n) (0).(H 1 , . . . , H n ) i j = σ∈Sn f ⊗ n (M, . . . , M ) i j σ(1) i σ(1) j σ(2) . . . i σ(n-1) j σ(n) i σ(n) j (H 1 ) j 1 i 1 . . . (H n ) jn in = n!f ⊗ n (M, . . . , M ) i j 1 i 1 j 2 . . . i n-1 jn in j (H 1 ) j 1 i 1 . . . (H n ) jn in
, where S n is the set of permutations of {1, . . . , n}.

For example, if f (z) = z n , we have f n = 1 and then f ⊗ n (M, . . . , M ) = I ⊗n+1 , but we have

σ∈Sn H σ(1) . . . H σ(n) = n!H 1 . . . H n .
If a function f : C → C is R-differentiable but not holomorphic, then f (M ) is well defined if M has no multiple eigenvalues. One can wonder if f (M ) is R-differentiable. The answer is yes, but to the author's knowledge, one does not have a nice expression of the derivative like we have in the holomorphic case. But we can use the fact that

f (M ) = λ eigenvalue of M f (λ)P M (λ),
where P M (λ) is the projection on the eigensubpace of M corresponding to the eigenvalue λ, parallelwise to all the other eigensubspaces of M .

We set M (t) = M + tH. Because of the continuity of eigenvalues, there exist continuous functions λ k (t) for 1 ≤ k ≤ dim(E), defined in a neighborhood of 0, such that for a given t, the λ k (t) are the eigenvalues of M (t). The projectors P M (t) (λ k (t)) are also continuous. As we will see below, the λ k (t) and P M (t) (λ k (t)) are analytic. So if we set F (t) = f (M (t)), we have

F (t) = dim(E) k=1 f (λ k (t))P M (t) (λ k (t)) 22 Thus if f is n times R-differentiable, we get F (n) (t) = dim(E) k=1 n h=0 n h d dt h f (λ k (t)) d dt n-h P M (t) (λ k (t)).
The derivatives of f (λ k (t)) can be expressed with the derivatives of f and λ k thanks to the Faa di Bruno formula. Now we look at the behaviour of λ k (t) and P M (t) (λ k (t)) near 0. Since the eigenvalues of M are different, we can set 2δ > 0 the minimum of the distance between two of them. So the balls B(λ k , δ) do not overlap (with λ k = λ k (0)). Because of the continuity of eigenvalues, there exists ε > 0 such that for |t| ≤ ε, we have λ k (t) ∈ B(λ k , δ) for all 1 ≤ k ≤ dim(E). Let us denote by u (λ) (δ) the characteristic function of B(λ, δ), which is holomorphic everywhere except on the boundary of B(λ, ε). Then we have, for t < ε,

λ k (t) = Tr(M (t)u (λ k ) (δ) (M (t))) P M (t) (λ k (t)) = u (λ k ) (δ) (M (t)).
To get the derivatives of these two guys, one can use Proposition 26, but using the fact that the derivative of zu

(λ k ) (δ) (z) is u (λ k ) (δ) (z) and Remark 28, we get λ (n) k (t) = Tr d dt n-1 P M (t) (λ k (t)
) H , so we only have to look at the derivatives of P M (t) (λ k (t)). Using Proposition 26, we get:

d dt n (P M (t) (λ k (t))) i j = n!u (λ k )⊗ (δ)n (M (t), . . . , M (t)) i j 1 i 1 j 2 . . . i n-1 jn in j H j 1 i 1 . . . H jn in .
If we compute that at t = 0, we can replace u

(λ k ) (δ) with u (λ k ) (δ ′ )
with δ ′ ≤ δ, since this two functions coincide on a neighborhood of the spectrum of M .

We have the following interesting fact: for all n the functions u

(δ)n simply converge to a limit we will denote by u (λ) n when δ tends to 0 (these functions are not defined on the whole space C n+1 , but for any single point, there are only finitely many bad δ's, and furthermore, for any (x 0 , . . . , x n ), the function δ → u

(λ) (δ)n (x 0 , . . . , x n ) is constant by parts). The function u (λ)
n is the symmetric function of n + 1 variables such that for all 0 ≤ m ≤ n + 1 and z 0 , . . . , z n-m = λ, we have

u (λ) n (λ, . . . , λ m times , z 0 , . . . , z-m) =      0 if m = 0 1 (m-1)! d dz m-1 n-m h=0 1 z-z h z=λ =(-1) (m-1) k 0 +...+k n-m =m-1 n-m h=0 1 (λ-z h ) 1+k h else. So we get d dt n (P M (t) (λ k (t))) i j = n!u (λ k (t))⊗ n (M (t), . . . , M (t)) i j 1 i 1 j 2 . . . i n-1 jn in j H j 1 i 1 . . . H jn in .
And finally, we have

d dt n (P M (t) (λ k (t)))=n! 1≤k 0 ,...,kn≤dim(E) u (λ k (t)) n (λ k 0 (t),...,λ kn (t))P M (t) (λ k 0 (t))HP M (t) (λ k 1 (t))H...HP M (t) (λ kn (t)) λ (n) k (t)=(n-1)! 1≤k 0 ,...,k n-1 ≤dim(E) u (λ k (t)) n-1 (λ k 0 (t),...,λ k n-1 (t)) Tr(P M (t) (λ k 0 (t))H...HP M (t) (λ k n-1 (t))H).
The following proposition shows how we can get i 1 =i 2 =... =i k f (λ i 1 , . . . , λ i k ) from the tensor f ⊗ (M, . . . , M ), where the λ i 's are the eigenvalues of M . This is a generalization of Tr(f (M )) = i f (λ i ).

Proposition 30 Let M be an endomorphism of a C-vector space E of finite dimension d. Let f be a function of k variables such that f ⊗ (M, . . . , M ) is well defined. We set λ 1 , . . . , λ d the eigenvalues of M (appearing with their multiplicities). Then we have the following:

1≤n 1 ,...,n k ≤d ∀l =l ′ ,n l =n l ′ f (λ n 1 , . . . , λ n k ) = k!f ⊗ (M, . . . , M ) i 1 j 1 . . . i k j k (Π ∧ k ) j 1 i 1 . . . j k i k , where (Π ∧ k ) j 1 i 1 . . . j k i k = 1 k! σ∈S k ǫ(σ)I j 1 i σ(1) I j 2 i σ(2) . . . I j k i σ(k) ,
with ǫ(σ) the signature of the permutation σ. The tensor (Π ∧ k ) is the one corresponding to the canonical projection from E ⊗k to the subspace E ∧k of antisymmetric tensors of order k.

Proof: Let σ ∈ S k . We set Ω(σ) the number of orbits of σ, and we denote by ω 0 , . . . , ω Ω(σ) these orbits. For 1 ≤ l ≤ k, we set 1 ≤ c(l) ≤ Ω(σ) the unique index such that l ∈ ω c(l) .

Then, using theorems 19 and 20, we have the following:

f ⊗ (M, . . . , M ) i 1 j 1 . . . i k j k I j 1 i σ(1) I j 2 i σ(2) . . . I j k i σ(k) = 1≤n 1 ,...,n Ω(σ) ≤d f (λ n c(1) , . . . , λ n c(k) ). Now, for 1 ≤ n 1 , . . . , n k ≤ d, we look at the number of times the term f (λ n 1 , . . . , λ n k ) appears when we use the formula above to compute σ∈S k ǫ(σ)f ⊗ (M, . . . , M ) i 1 j 1 . . . i k j k I j 1 i σ(1) I j 2 i σ(2) . . . I j k i σ(k) .

This term appears once for each σ such that l → n l is constant on all the orbits of σ, with the prefactor ǫ(σ). The set of such permutations is in fact a subgroup of S k , more precisely the one of permutations which stabilize the equivalence classes of the relation l ∼ l ′ ⇔ n l = n l ′ .

If there exist two different indexes p and q such that n p and n q are equal, then the above-mentionned subgroup contains the transposition (p, q) which has signature -1, and thus, since ǫ is a group morphism, half of the elements of the subgroup has signature 1 and the other half has signature -1, so the sum of the signatures is 0.

If all the n l are different, then the subgroup is just the identity, so our term only appears once.

Hence we have σ∈S k ǫ(σ)f ⊗ (M, . . . , M ) i 1 j 1 . . . i k j k I j 1 i σ(1) I j 2 i σ(2) . . . I j k i σ(k) = The eigenvalues of f ∧ (M, . . . , M ) are 1 k! σ∈S k f (λ n σ(1) , . . . , λ n σ(k) ) for 1 ≤ n 1 < n 2 < . . . < n k ≤ d (the corresponding eigenvector is v n 1 ∧ . . . ∧ v n k if M is diagonalizable and if v 1 , . . . , v d is a basis of eigenvectors of M , such that v n is an eigenvector of M for the eigenvalue λ n ).

One could want to extend the definition of f ∧ (M, . . . , M ) to some cases in which f ⊗ (M, . . . , M ) is not well defined. For example, for f (x, y) = 1 (x-y) 2 , f ∧ (M, M ) should have a sense if M only has simple eigenvalues, whereas f ⊗ (M, M ) has no sense, whatever M is. But the minimal conditions we should put on f and the M l 's to extend the definition of f ∧ (M 1 , . . . , M k ) are not clear.

If the fact that f ⊗ (M 1 , . . . , M k ) is a tensor and not a matrix is disturbing, there could be a way to define a kind of f (M 1 , . . . , M k ) which would be a matrix. One way to get a matrix from the tensor f ⊗ (M 1 , . . . , M k ) is to use contractions. Assume that one can write f (x 1 , . . . , x k ) = a n 1 ,...,n k x n 1 1 . . . x n k k , where we have |a n 1 ,...,n k |ρ n 1 1 . . . ρ n k k < ∞ for some ρ 1 , . . . , ρ k > 0. Assume that for every 1 ≤ l ≤ k, we have sup λ eigenvalue of M l |λ| < ρ l . Then the sum:

a n 1 ,...,n k (M n 1 1 M n 2 2 . . . M n k k ) i j
is convergent, and this matrix is exactly

f ⊗ (M 1 , . . . , M k ) i i 1 i 1 i 2 . . . i k-2 i k-1 i k-1 j .
It is what we expect f (M 1 , . . . , M k ) to be when all the M l commute. But if they do not commute, this may introduce a dissymetry. It does still work well if f (x 1 , . . . , x k ) = k l=1 f l (x l ). But in the simple example f (x, y) = (x + y) 2 , we have f ⊗ (A, B) i k k j = (A 2 + B 2 + 2AB) i j and not ((A + B) 2 ) i j as one could want.

If f (x, y) = 1 x+y , and provided none of the eigenvalues of A is the opposite of an eigenvalue of B, it gives you the unique matrix M such that AM + M B = I, and this matrix is not (A + B) -1 in general.

,

  the λ lm are the eigenvalues of M l and the r lm are the sizes of the Jordan blocks of M l . Let P be a polynomial on k variables.Let us choose the values of the indexes i 1 , . . . , i k and j 1 , . . . , j k . Then for each 1 ≤ l ≤ k, there exist1 ≤ c l , d l ≤ b l such that c l -1 m=1 r lm < i l ≤ c l m=1 r lm and d l -1 m=1 r lm < j l ≤ d l m=1 r lm . Then the coefficient P ⊗ (M 1 , .

  kq l=1 ∂ j l l R(λ q1m 1 , . . . , λ qkqm kq ) = 0 and then by Proposition 6, one has R ⊗ (M q1 , . . . , M qkq ), and hence Q( Mq ) = 0 as stated. Now let P be a suitable interpolation polynomial of g, i.e. such that µ 1p 1 , . . . , µ rpr ),

  λ 11m 11 , . . . , λ rkrm rkr ) = λ 11m 11 , . . . , λ rkrm rkr ),

1≤n 1 ,

 1 ...,n k ≤d ∀l =l ′ ,n l =n l ′ f (λ n 1 , . . . , λ n k ), as wanted.A classical example is given by taking k = d, and f (x 1 , . . . , x d ) = x 1 . . . x d , in which case we can getdet(M ) = a 1 +2a 2 +...+da d =d (-1) d-a 1 -a 2 -...-a d a 1 !a 2 ! . . . a d !1 a 1 2 a 2 . . . d a d Tr(M ) a 1 Tr(M 2 ) a 2 . . . Tr(M d ) a d .The restriction to E ∧k of Π ∧ k f ⊗ (M, . . . , M ), where Π ∧ k and f ⊗ (M, . . . , M ) are seen as endomorphisms of E ⊗ k, can also be itself interesting (and not only its trace). Let us denote it by f ∧ (M, . . . , M ).Indeed, since E ∧k has dimension d k , f ∧ (M, . . . , M ) lives in a space of dimension d k 2 , whereas f ⊗ (M, . . . , M ) lives in a space of much greater dimension d 2k .