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Digital Repository of Scientific Institutes (RCIN) was established and is being developed mainly as an outcome of two projects acquired and jointly accomplished by a group of Polish scientific institutes.

-matrices and operations on matrices; the algorithm of gauss and some of its applications; linear operators in an n-dimensional vector space; the characteristic polynomial and the minimal polynomial of a matrix; functions of matrices; equivalent transformations of polynomial matrices; analytic theory of elementary divisors; the structure of a linear operator in an n-dimensional space; matrix equations; linear operators in a unitary space; quadratic and hermitian forms-

42 p. 28 cm

A Supplementary Memoir on the Theory of Matrices. Cayley, A Proceedings of the Royal Society of London (1854-1905). 1865-01-01. 14:543–543

Digital Repository of Scientific Institutes (RCIN) was established and is being developed mainly as an outcome of two projects acquired and jointly accomplished by a group of Polish scientific institutes.

Okamoto has obtained a sequence of $\tau$-functions for the \PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multi-dimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter $N$, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the \PVI theory. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter $a$ a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter $a$ an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the $\tau$-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter $a=0$; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

Multi-conductor ribbon cables using closely coupled dielectric-coated conductors are now used extensively to interconnect electronic subsystems. General techniques involving solution of the multiconductor transmission line equations can be used to predict unwanted interference or crosstalk in the cable. Inherent in these techniques is knowledge of the per-unit-length transmission line capacitance and inductance matrices of the ribbon cable. A computer implemented numerical technique for determining these matrices for ribbon cables is described. The per-unit-length charge distributions on the conductor and dielectric surfaces are described by Fourier series with the potential and displacement vector equations at these surfaces enforced as boundary conditions. A metric inversion techniques is used to develop elements of the generalized and transmission line capacitance matrices. Computer program GETCAP was written in a user-oriented format. The user simply describes the physical characteristics and GETCAP computes the per-unit-length capacitance matrices for the ribbon cable described.

Multi-conductor ribbon cables using closely coupled dielectric-coated conductors are now used extensively to interconnect electronic subsystems. General techniques involving solution of the multiconductor transmission line equations can be used to predict unwanted interference or crosstalk in the cable. Inherent in these techniques is knowledge of the per-unit-length transmission line capacitance and inductance matrices of the ribbon cable. A computer implemented numerical technique for determining these matrices for ribbon cables is described. The per-unit-length charge distributions on the conductor and dielectric surfaces are described by Fourier series with the potential and displacement vector equations at these surfaces enforced as boundary conditions. A metric inversion techniques is used to develop elements of the generalized and transmission line capacitance matrices. Computer program GETCAP was written in a user-oriented format. The user simply describes the physical characteristics and GETCAP computes the per-unit-length capacitance matrices for the ribbon cable described.

The {\it theory unifying spin and charges and predicting families}, proposed by N.S.M.B., predicts at the low energy regime two (in the mixing matrix elements decoupled) groups of four families. There are two kinds of contributions to mass matrices in this theory. One kind distinguishes on the tree level only among the members of one family, that is among the $u$-quark, $d$-quark, neutrino and electron, the left and right handed, while the other kind distinguishes only among the families. Mass matrices for $d$-quarks and electrons are on the tree level correspondingly strongly correlated and so are mass matrices for $u$-quarks and neutrinos, up to the term, the Majorana term, which is nonzero only for right handed neutrinos. Beyond the tree level both kinds of contributions start to contribute coherently and it is expected that a detailed study of properties of mass matrices beyond the tree level explains drastic differences in masses and mixing matrices between quarks and leptons. We report in this paper on analysis of one loop corrections to the tree level fermion masses and mixing matrices. Loop diagrams are mediated by the gauge bosons and the two kinds of scalar fields. A detailed numerical analysis of fermion masses and mixing, including neutrinos, within this scenario is in progress and preparation.

The {\it theory unifying spin and charges and predicting families}, proposed by N.S.M.B., predicts at the low energy regime two (in the mixing matrix elements decoupled) groups of four families. There are two kinds of contributions to mass matrices in this theory. One kind distinguishes on the tree level only among the members of one family, that is among the $u$-quark, $d$-quark, neutrino and electron, the left and right handed, while the other kind distinguishes only among the families. Mass matrices for $d$-quarks and electrons are on the tree level correspondingly strongly correlated and so are mass matrices for $u$-quarks and neutrinos, up to the term, the Majorana term, which is nonzero only for right handed neutrinos. Beyond the tree level both kinds of contributions start to contribute coherently and it is expected that a detailed study of properties of mass matrices beyond the tree level explains drastic differences in masses and mixing matrices between quarks and leptons. We report in this paper on analysis of one loop corrections to the tree level fermion masses and mixing matrices. Loop diagrams are mediated by the gauge bosons and the two kinds of scalar fields. A detailed numerical analysis of fermion masses and mixing, including neutrinos, within this scenario is in progress and preparation.

The {\it theory unifying spin and charges and predicting families}, proposed by N.S.M.B., predicts at the low energy regime two (in the mixing matrix elements decoupled) groups of four families. There are two kinds of contributions to mass matrices in this theory. One kind distinguishes on the tree level only among the members of one family, that is among the $u$-quark, $d$-quark, neutrino and electron, the left and right handed, while the other kind distinguishes only among the families. Mass matrices for $d$-quarks and electrons are on the tree level correspondingly strongly correlated and so are mass matrices for $u$-quarks and neutrinos, up to the term, the Majorana term, which is nonzero only for right handed neutrinos. Beyond the tree level both kinds of contributions start to contribute coherently and it is expected that a detailed study of properties of mass matrices beyond the tree level explains drastic differences in masses and mixing matrices between quarks and leptons. We report in this paper on analysis of one loop corrections to the tree level fermion masses and mixing matrices. Loop diagrams are mediated by the gauge bosons and the two kinds of scalar fields. A detailed numerical analysis of fermion masses and mixing, including neutrinos, within this scenario is in progress and preparation.

"On the Theory of Matrices" is an article from American Journal of Mathematics, Volume 12 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2369849

"A Memoir on the Theory of Matrices" is an article from Philosophical Transactions of the Royal Society of London, Volume 148 . View more articles from Philosophical Transactions of the Royal Society of London . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-108649

The matrix calculus is widely applied nowadays in various branches of mathematics, mechanics, theoretical physics, theoretical electrical engineer­ ing, etc. However, neither in the Soviet nor the foreign literature is there a book that gives a sufficiently complete account of the problems of matrix theory and of its diverse applications. The present book is an attempt to fill this gap in the mathematical literature. The book is based on lecture courses on the theory of matrices and its applications that the author has given several times in the course of the last seventeen years at the Universities of Moscow and Tiflis and at the Moscow Institute of Physical Technology. The book is meant not only for mathematicians (undergraduates and research students) but also for specialists in allied fields (physics, engi­ neering) who are interested in mathematics and its applications. Therefore the author has endeavoured to make his account of the material as accessible as possible, assuming only that the reader is acquainted with the theory of determinants and with the usual course of higher mathematic? within the programme of higher technical education.

Bibliography: p. 205-248

Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over finite filed; III.A Law of Large Numbers for the characters of GL_n(k) over finite field k; IV.An outline of construction of factor representations of the group GLB(F_q).

-complex symmetric, skew-symmetric and orthogonal matrices; singular pencils of matrices; matrices with non-negative elements; applications of the theory of matrices to the investigation of systems of linear differential equations; the problem of routh-hurwitz and related questions-

A Memoir on the Theory of Matrices. Cayley, A Proceedings of the Royal Society of London (1854-1905). 1857-01-01. 9:100–101

A Memoir on the Theory of Matrices. Cayley, A Proceedings of the Royal Society of London (1854-1905). 1857-01-01. 9:100–101

In the first half of this text we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch & Silbermann and Lindner) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang. We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator $A$. In the second half of this text we study bounded linear operators on the generalised sequence space $\ell^p(\Z^N,U)$, where $p\in [1,\infty]$ and $U$ is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator $A$ is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of $p=1$ and $\infty$. Our tools in this study are the results from the first half of the text and an exploitation of the partial duality between $\ell^1$ and $\ell^\infty$. Results in this second half of the text include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schr\"odinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on $\R^N$.

Combinatorial formulas expressing cyclic rook polynomials and cyclic permanents of rectangular matrices in terms of expansions along rows are presented

This document summarizes 46 research papers which have either appeared or will appear in major referred publications. Highlights of some of the results include: (1) Convenient stability criteria for difference approximations to hyperbolic initial boundary value problems have been obtained. The new criteria are given in terms of the boundary conditions and are independent of the basic scheme. (2) The known stability condition for the multi-dimensional Lax-Wendroff scheme has been improved. (3) Norm properties of C-numerical radii were studied. In particular, multiplicativity factors were obtained for the case where C is a normal matrix. Also, the following computer codes written for the ILLIAC IV were completed: (1) a program to determine the rank and a maximal nonvanishing subdeterminant of an integral m X n matrix; (2) a program to determine the null space of an integral m X n matrix; and (3) a program to determine all the rational solutions of an arbitrary integral linear system of equations. (Author)

This document summarizes 46 research papers which have either appeared or will appear in major referred publications. Highlights of some of the results include: (1) Convenient stability criteria for difference approximations to hyperbolic initial boundary value problems have been obtained. The new criteria are given in terms of the boundary conditions and are independent of the basic scheme. (2) The known stability condition for the multi-dimensional Lax-Wendroff scheme has been improved. (3) Norm properties of C-numerical radii were studied. In particular, multiplicativity factors were obtained for the case where C is a normal matrix. Also, the following computer codes written for the ILLIAC IV were completed: (1) a program to determine the rank and a maximal nonvanishing subdeterminant of an integral m X n matrix; (2) a program to determine the null space of an integral m X n matrix; and (3) a program to determine all the rational solutions of an arbitrary integral linear system of equations. (Author)

Since the theory of matrices and vectors is one of the main concepts of higher mathematics, they play a leading role in solving problems in other disciplines. In this article, we present that one of them is used to solve economic issues. Let's say that a car factory produces four different brands of cars in a month. If the following main characteristics of car production are given in this table by a matrix

The conductance of a normal-metal mesoscopic system in proximity to superconducting electrode(s) is calculated. The normal-metal part may have a general geometry, and is described as a ``circuit'' with ``leads'' and ``junctions''. The junctions are each ascribed a scattering matrix which is averaged over the circular orthogonal ensemble, using recently-developed techniques. The results for the electrical conductance reproduce and extend Nazarov's circuit theory, thus bridging between the scattering and the bulk approaches. The method is also applied to the heat conductance.

The conductance of a normal-metal mesoscopic system in proximity to superconducting electrode(s) is calculated. The normal-metal part may have a general geometry, and is described as a ``circuit'' with ``leads'' and ``junctions''. The junctions are each ascribed a scattering matrix which is averaged over the circular orthogonal ensemble, using recently-developed techniques. The results for the electrical conductance reproduce and extend Nazarov's circuit theory, thus bridging between the scattering and the bulk approaches. The method is also applied to the heat conductance.

A Supplementary Memoir on the Theory of Matrices. Cayley, A Philosophical Transactions of the Royal Society of London (1776-1886). 1866-01-01. 156:25–35

First reprinting, 1963, Johnson Reprint Corporation

"On Certain Identities in the Theory of Matrices" is an article from American Journal of Mathematics, Volume 13 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2369812

"On Certain Identities in the Theory of Matrices" is an article from American Journal of Mathematics, Volume 13 . View more articles from American Journal of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2369812

We announce three results in the theory of Jacobi matrices and Schr\"odinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schr\"odinger operator $-\f{d^2}{dx^2} +V(x)$ on $L^2 (0,\infty)$ with $V\in L^2 (0,\infty)$ and $u(0)=0$ boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szeg\H{o} asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.

I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (`Laplacian') matrices, line graphs, and very strong regularity.

I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (`Laplacian') matrices, line graphs, and very strong regularity.

This paper deals with the computation of the rank and of some integer Smith forms of a series of sparse matrices arising in algebraic K-theory. The number of non zero entries in the considered matrices ranges from 8 to 37 millions. The largest rank computation took more than 35 days on 50 processors. We report on the actual algorithms we used to build the matrices, their link to the motivic cohomology and the linear algebra and parallelizations required to perform such huge computations. In particular, these results are part of the first computation of the cohomology of the linear group GL_7(Z).

"A Supplementary Memoir on the Theory of Matrices. [Abstract]" is an article from Proceedings of the Royal Society of London, Volume 14 . View more articles from Proceedings of the Royal Society of London . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-112178

With $ $ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,\mu) := < \prod_{l=1}^N \chi_{(0,\infty)\backslash I}^{(l)} (\lambda - \lambda_l)^\mu>$ for $I = (0,s)$ and $I = (s,\infty)$, where $\chi_I^{(l)} = 1$ for $ \lambda_l \in I$ and $\chi_I^{(l)} = 0$ otherwise. Using Okamoto's development of the theory of the Painlev\'e V equation, it is shown that $\tilde{E}_N(I;a,\mu)$ is a $\tau$-function associated with the Hamiltonian therein, and so can be characterised as the solution of a certain second order second degree differential equation, or in terms of the solution of certain difference equations. The cases $\mu = 0$ and $\mu = 2$ are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case $I = (s,\infty)$, $\tilde{E}_N(I;a,\mu)$ is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of $\tilde{E}_N(I;a,\mu)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity $E^{\rm hard}((0,s);a,\mu)$ can be evaluated as a $\tau$-function associated with the Hamiltonian in Okamoto's theory of the Painlev\'e III equation.

With $ $ denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is $ \tilde{E}_N(I;a,\mu) := < \prod_{l=1}^N \chi_{(0,\infty)\backslash I}^{(l)} (\lambda - \lambda_l)^\mu>$ for $I = (0,s)$ and $I = (s,\infty)$, where $\chi_I^{(l)} = 1$ for $ \lambda_l \in I$ and $\chi_I^{(l)} = 0$ otherwise. Using Okamoto's development of the theory of the Painlev\'e V equation, it is shown that $\tilde{E}_N(I;a,\mu)$ is a $\tau$-function associated with the Hamiltonian therein, and so can be characterised as the solution of a certain second order second degree differential equation, or in terms of the solution of certain difference equations. The cases $\mu = 0$ and $\mu = 2$ are of particular interest as they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue. In the case $I = (s,\infty)$, $\tilde{E}_N(I;a,\mu)$ is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard edge and soft edge scaled limits of $\tilde{E}_N(I;a,\mu)$. In particular, in the hard edge scaled limit it is shown that the limiting quantity $E^{\rm hard}((0,s);a,\mu)$ can be evaluated as a $\tau$-function associated with the Hamiltonian in Okamoto's theory of the Painlev\'e III equation.

"A Memoir on the Theory of Matrices. [Abstract]" is an article from Proceedings of the Royal Society of London, Volume 9 . View more articles from Proceedings of the Royal Society of London . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-111451

"A Memoir on the Theory of Matrices. [Abstract]" is an article from Proceedings of the Royal Society of London, Volume 9 . View more articles from Proceedings of the Royal Society of London . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-111451

We characterize the cone of GL-equivariant Betti tables of Cohen-Macaulay modules of codimension 1, up to rational multiple, over the coordinate ring of square matrices. This result serves as the base case for `Boij-S\"oderberg theory for Grassmannians', with the goal of characterizing the cones of GL_k-equivariant Betti tables of modules over the coordinate ring of k x n matrices, and, dually, cohomology tables of vector bundles on the Grassmannian Gr(k, C^n). The proof uses Hall's Theorem on perfect matchings in bipartite graphs to compute the extremal rays of the cone, and constructs the corresponding equivariant free resolutions by applying Weyman's geometric technique to certain graded pure complexes of Eisenbud-Fl{\o}ystad-Weyman.

It is shown that the world-line can be eliminated in the matrix quantum mechanics conjectured by Banks, Fischler, Shenker and Susskind to describe the light-cone physics of M theory. The resulting matrix model has a form that suggests origins in the reduction to a point of a Yang-Mills theory. The reduction of the Nishino-Sezgin 10+2 dimensional supersymmetric Yang-Mills theory to a point gives a matrix model with the appropriate features: Lorentz invariance in 9+1 dimensions, supersymmetry, and the correct number of physical degrees of freedom.

It is shown that the world-line can be eliminated in the matrix quantum mechanics conjectured by Banks, Fischler, Shenker and Susskind to describe the light-cone physics of M theory. The resulting matrix model has a form that suggests origins in the reduction to a point of a Yang-Mills theory. The reduction of the Nishino-Sezgin 10+2 dimensional supersymmetric Yang-Mills theory to a point gives a matrix model with the appropriate features: Lorentz invariance in 9+1 dimensions, supersymmetry, and the correct number of physical degrees of freedom.

This document is designed to provide a concise introduction to the theory of generalized inverses of matrices that is accessible to undergraduate mathematics majors. The approach used is to: (1) develop the material in terms of full-rank factorizations and to relegate all discussions using eigenvalues and eigenvectors to exercises, and (2) include an appendix of hints for exercises. In addition, the Moore-Penrose inverse of a matrix is introduced and its use in characterizing particular solutions to systems of equations is immediately explored before many of its algebraic properties are established. This is done to provide some motivation for considering generalized inverses before developing the algebraic theory. This material was originally assembled as lecture notes for senior seminars in mathematics at the University of Tennessee, and requires a knowledge of basic matrix theory as found in many current introductory texts. It is noted that it may be helpful to have a standard linear algebra textbook for reference, as fundamental definitions and concepts are used without the detailed discussion that would be included in a self-contained work. (MP)

288 p. 25 cm

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of $\tilde{E}_N(\lambda;a) := \Big < \prod_{l=1}^N \chi_{(-\infty, \lambda]}^{(l)} (\lambda - \lambda_l)^a \Big>$, where $ \chi_{(-\infty, \lambda]}^{(l)} = 1$ for $\lambda_l \in (-\infty, \lambda]$ and $ \chi_{(-\infty, \lambda]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $F_N(\lambda;a) := \Big < \prod_{l=1}^N (\lambda - \lambda_l)^a \Big >$. Of particular interest are $\tilde{E}_N(\lambda;2)$ and $F_N(\lambda;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\tau$-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities $\tilde{E}_N(\lambda;a)$ and $F_N(\lambda;a)$ for the other classical matrix ensembles.

Book Source: Digital Library of India Item 2015.4951 dc.contributor.author: Turnbull.h.w dc.date.accessioned: 2015-06-19T14:57:51Z dc.date.available: 2015-06-19T14:57:51Z dc.date.digitalpublicationdate: 2005-06-15 dc.date.citation: 1929 dc.identifier: Librarian, Bits-Pilani dc.identifier.barcode: 99999990149085 dc.identifier.origpath: /data1/upload/0004/182 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/4951 dc.description.scanningcentre: C-DAC, Noida dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 360 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Digital Library Of India dc.publisher: Blackie And Son Limited,london dc.rights: Copyright Permitted dc.source.library: Central Library, Bits- Pilani dc.subject.classification: Generalities dc.subject.keywords: Transposed dc.subject.keywords: Permutation dc.subject.keywords: Convolutions dc.title: The Theory Of Determinants Matrices And Invariants

A Memoir on the Theory of Matrices. Cayley, A Philosophical Transactions of the Royal Society of London (1776-1886). 1858-01-01. 148:17–37

A Memoir on the Theory of Matrices. Cayley, A Philosophical Transactions of the Royal Society of London (1776-1886). 1858-01-01. 148:17–37