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An algebraic approach to nonorthogonal general joint block diagonalization. (English) Zbl 1354.15008

Summary: The exact/approximate nonorthogonal general joint block diagonalization (nogjbd) problem of a given real matrix set \(\mathcal{A}=\{A_i\}_{i=1}^m\) is to find a nonsingular matrix \(W\in\mathbb{R}^{n\times n}\) (diagonalizer) such that \(W^T A_i W\) for \(i=1,2,\dots, m\) are all exactly/approximately block diagonal matrices with the same diagonal block structure and with as many diagonal blocks as possible. In this paper, we show that a solution to the exact/approximate nogjbd problem can be obtained by finding the exact/approximate solutions to the system of linear equations \(A_iZ=Z^TA_i\) for \(i=1,\dots, m\), followed by a block diagonalization of \(Z\) via similarity transformation. A necessary and sufficient condition for the equivalence of the solutions to the exact nogjbd problem is established. Two numerical methods are proposed to solve the nogjbd problem, and numerical examples are presented to show the merits of the proposed methods.

MSC:

15A21 Canonical forms, reductions, classification
15A69 Multilinear algebra, tensor calculus
65F30 Other matrix algorithms (MSC2010)

Software:

Tensorlab
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Full Text: DOI arXiv

References:

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