2007.99: MATLAB Toolbox for Classical Matrix Groups

2007.99: David P. Jagger (2003) MATLAB Toolbox for Classical Matrix Groups. Masters thesis, University of Manchester.

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Abstract

We consider structured matrix groups arising in the context of nondegenerate bilinear or sesquilinear forms on $\Rn$ or $\Cn$, which remain invariant under similarities by matrices in their associated automorphism group. We develop a \M toolbox for generating random matrices from these automorphism groups with, wherever possible, prescribed condition numbers. The matrix groups considered are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudo-orthogonal, and pseudo-unitary groups.

We outline all necessary background theory before presenting a self-contained treatment of each group, outlining some applications of the group and deriving an algorithm for their random generation. We first focus our attention on the groups for which a structured SVD or CSD is available, and show that this allows for precise control of the condition number via numerically stable algorithms. We then consider the groups which lack such a decomposition, where we construct matrices via products of generalized $\Ga$-reflectors. We perform tests which model the behaviour of the condition number in these cases, allowing for its approximate control, and finally we consider the effect of rounding errors on the resultant matrices.

The implementation in \M of these algorithms is hoped to be beneficial for researchers developing structure-preserving algorithms for structured problems.

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