Functions of Matrix Argument

Abstract

Particular cases of a H-function with matrix argument are available for real as well as for complex matrices. For the general H-function only a class of functions is available analogous to the scalar variable H-function. Real-valued scalar functions of matrix argument is developed when the argument matrix is a real symmetric positive definite matrix or for hermitian positive definite matrices. We consider only real matrices here.

We will use the standard notations to denote matrices. The transpose of a matrix X = (x _ij ) will be denoted by X ^′ and trace of X by tr(X) = sum of the eigenvalues = sum of the leading diagonal elements. Determinant of X will be denoted by | X | , a null matrix by a big O and an identity matrix by I = I _n . A diagonal matrix will be written as diag(λ₁, …, λ _p ) where λ₁, …, λ _p are the diagonal elements. X > 0 will mean the real symmetric matrix X = X ^′ is positive definite. Definiteness is defined only for symmetric matrices when real and hermitian matrices when complex, X ≥ 0 (positive semidefinite), X < 0 (negative definite), X ≤ 0 (negative semidefinite). A matrix which does not fall in the categories X > 0, X ≥ 0, X < 0, X ≤ 0 is called indefinite. ∫ _X f(X)dX means the integral over X. ${\int \nolimits \nolimits }_{A}^{B}f(X)\mathrm{d}X$ means the integral over 0 < A < X < B, that is, X = X ^′ > 0, A = A ^′ > 0, B = B ^′ > 0, X − A > 0, B − X > 0 and the integral is taken over all such X.