Computing the inverse matrix hyperbolic sine
by
João R. Cardoso
Instituto Superior de Engenharia de Coimbra - Portugal
Coauthors: Fátima Silva Leite (Universidade de Coimbra)

For any real or complex square matrix X, the hyperbolic sine, naturally defined as sinhX:=(e^X-e^-X)/2, is a primary matrix function. Any matrix X which satisfies the matrix equation sinhX=A, for a given A, is called an inverse matrix hyperbolic sine of A. This equation may have solutions or not; it depends on the eigenvalues of A. Conditions on the spectrum of A will be presented for the case when A is real and also for the complex case. When the equation has some solution, in general there is an infinity. Among these we are particularly interested in the one with eigenvalues lying on the strip { z in \dC: -\pi/2 < Im(z) < \pi/2}. We will show that there is a unique such solution, which we call the principal inverse matrix hyperbolic sine of A.

Our interest on computing the inverse matrix hyperbolic sine was motivated by the work of Crouch and Bloch (1996), where the matrix equation XQ^T-QX^T=M appears associated with the generalized rigid body equations. In this case Q is orthogonal and M is skew-symmetric. It turns out that X=(e^{sinh-1\fracM2})Q is a solution of that matrix equation. As far as we know, the problem of computing inverse matrix hyperbolic (or trigonometric) functions has not been paid much attention in the literature. However, recent developments on computing matrix logarithms can be carefuly manipulated to produce an algorithm for computing the principal inverse matrix hyperbolic sine. This algorithm may be slightly modified to adjust to the computation of other inverse matrix hyperbolic (or trigonometric) functions. The algorithm presented here holds for matrices with no eigenvalues on {\alphai: \alpha in \dR, \alpha > 1}. The numerical features of this algorithm will be analyzed along with some numerical tests. When A is a matrix satisfying A^TP=PA, with P invertible, slight modifications on that algorithm turn it into a structure preserving method. Also, since in general skew-symmetric matrices do not satisfy the spectral condition of the algorithm, an alternative procedure is presented for this case.

Key-words: primary matrix function, inverse matrix hyperbolic sine, matrix exponentials and logarithms, Pad\' e approximants.

Date received: February 1, 2000