Matrix functions with potential applications have a major role in science and engineering. One of the fundamental matrix functions, which is particularly important due to its connections with certain matrix differential equations and other special matrix functions, is the matrix gamma function. This research article is focused on the numerical computation of this function. Well-known techniques for the scalar gamma function, such as Lanczos and Spouge methods, are carefully extended to the matrix case. This extension raises many challenging issues and several strategies used in the computation of matrix functions, like Schur decomposition and block Parlett recurrences, need to be incorporated to turn the methods more effective. We also propose a third technique based on the reciprocal gamma function that is shown to be competitive with the other two methods in terms of accuracy, with the advantage of being rich in matrix multiplications. Strengths and weaknesses of the proposed methods are illustrated with a set of numerical examples. Bounds for truncation errors and other bounds related with the matrix gamma function will be discussed as well.Since the reciprocal gamma function, here denoted by ∆(z) := 1 Γ(z) , is an entire function, for any matrix A ∈ C n×n , the n × n matrix ∆(A) = (Γ(A)) −1 is a well defined matrix. Furthermore, if A does not have any non-positive integer eigenvalue, then A + mI is invertible, for all integer m ≥ 0, and one gets the following formula [28], which uses the Pochhammer notation: (A) m = A(A + I) . . . (A + (m − 1)I) = Γ(A + mI)∆(A), m ≥ 1 and (A) 0 = I. An alternative definition of the matrix gamma function, as a limit of a sequence of matrices, is provided in [28]: Γ(A) = lim m→∞ (m − 1)! [(A) m ] −1 m A .We shall note that many definitions of the scalar gamma function (see, for instance, [1, Ch. 6]) may be easily extended to the matrix case. The matrix gamma function has connections with other special functions, which in turn play an important role to solving certain matrix differential equations; see [28] and the references therein. Two of those special functions are the matrix beta and Bessel functions. If the matrices A, B ∈ C n×n satisfy the spectral conditions [16] Re(z) > −1, ∀z ∈ σ(A), Re(z) < 1, ∀z ∈ σ(B), then 1 −1 (1 + t) A (1 − t) B dt = 2 A+I B(A + I, B + I) 2 B , where B(A, B) is the Beta matrix function [28], defined by

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If A has no eigenvalues on the closed negative real axis, and B is arbitrary square complex, the matrix-matrix exponentiation is defined as A B := e log(A)B . This function arises, for instance, in Von Newmann's quantum-mechanical entropy, which in turn finds applications in other areas of science and engineering. Since in general A and B do not commute, this bivariate matrix function may not be a primary matrix function as commonly defined, which raises many challenging issues. In this paper, we revisit this function and derive new related results. Particular emphasis is given to its Fréchet derivative and conditioning. We present a general result on the Fréchet derivative of bivariate matrix functions with applications not only to the matrix-matrix exponentiation but also to other functions, such as the second order Fréchet derivatives and some iteration functions arising in matrix iterative methods. The numerical computation of the Fréchet derivative is discussed and an algorithm for computing the relative condition number of A B is proposed. Some numerical experiments are included.

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