Jacobians of Matrix Transformations and Functions of Matrix Argument

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1 Jacobians of Matrix Transformations and Functions of Matrix Argument A. M. Mathai Department of Mathematics & Statistics, McGill University World Scientific Singapore *New Jersey London Hong Kong

2 Contents 1. Jacobians of matrix transformations Introduction Vector and matrix derivatives The derivative of a matrix with respect to a scalar The derivative of a scalar function of a vector with respect to the vector The derivative of a scalar function of a matrix with respect to the matrix The derivative of a matrix with respect to an element of the same matrix 14 Exercises Jacobians of linear matrix transformations Jacobians of vector transformations Linear matrix transformations Function of a function, linear case 31 Exercises Nonlinear transformations: one matrix case Nonlinear vector transformations Smooth nonlinear transformations Function of a function Jacobians in terms of eigenvalues 66 Exercises Jacobians in orthogonal and related transformations Introduction Transformations of one matrix to components of two matrices Decomposition with one matrix being triangular or diagonal Decomposition with one matrix being skew symmetric 99 Exercises Decomposition with one matrix being orthogonal or and element of the Stiefel manifold 115 IX

3 Exercises Singular value decomposition 137 _ Square matrices of full rank Rectangular matrices of full rank 142 Exercises Simultaneous transformations involving many matrices Transformations involving two matrices Transformations involving many matrices 153 Exercises Transformations involving submatrices 159 Exercises Jacobians in the complex case Introduction Jacobians of linear transformations in the complex case 175 Exercises Smooth nonlinear transformations in the complex case 187 Exercises Transformations involving determinants and diagonal matrices 199 Exercises Transformations involving eigenvalues and unitary matrices Introduction Transformations involving skew hermitian matrices A direct evaluation of the integral 228 Exercises Transformations involving unitary matrices Semiunitary transformations 240 Exercises Transformations involving singular values Square matrices of full rank Rectangular matrices of full rank 250 Exercises Some special functions of matrix argument Introduction Elementary functions of matrix argument Matrix-variate real gamma density 254

4 5.1.2 Laplace transform Functions of matrix argument through Laplace transform Matrix-variate real beta density Functions of matrix argument through zonal polynomials The Weyl fractional integral and fractional derivative The M-transform Matrix-variate Dirichlet integrals and Dirichlet distributions Real matrix-variate Liouville function and Liouville distribution Some applications 279 Exercises Hypergeometric functions of matrix argument Hypergeometric functions of matrix argument through Laplace transform Hypergeometric functions through zonal polynomials Bessel function of matrix argument Hypergeometric function of matrix argument through M-transform Whittaker function of matrix argument Meijer's G- and Fox's H-functions of matrix argument through M-transform Some applications 303 Exercises Appell's and Humbert's functions Appell's functions of matrix arguments Humbert's functions of matrix arguments Humbert's functions as limiting forms 317 Exercises Kampe' de Fe'riet's functions of matrix arguments Definition Some integral representations Some limiting forms Some reduction formulae An application in astrophysics problems 331 Exercises Lauricella functions of matrix arguments 333

5 5.5.1 Lauricella function /A of matrix arguments Lauricella function fs of matrix arguments Lauricella function fc of matrix arguments Lauricella function fo of matrix arguments Some mixed results Some applications to statistical distributions 347 (a) Distribution of a linear form 347 (b) Distribution of a determinant 352 (c) Generalized quadratic forms 353 Exercises Functions of matrix argument in the complex case Introduction Some elementary functions in the complex case Laplace transform in the complex case Zonal polynomials in the complex case The M-transform 367 Exercises Hypergeometric functions of matrix argument: complex case 369 Exercises G- and H-functions of matrix argument: complex case 375 Exercises Functions of several matrix arguments: complex case Dirichlet functions: complex case Appell's functions: complex case Series forms: complex case Humbert's functions: complex case Kampe" de Fe"riet's functions: complex case Lauricella functions: complex case 396 Exercises 399 Appendix 401 Bibliography 411 Glossary of symbols 421 Author index 425 Subject index 429

ELEMENTARY MATRIX ALGEBRA

ELEMENTARY MATRIX ALGEBRA Third Edition FRANZ E. HOHN DOVER PUBLICATIONS, INC. Mineola, New York CONTENTS CHAPTER \ Introduction to Matrix Algebra 1.1 Matrices 1 1.2 Equality of Matrices 2 13 Addition