The matrix sign decomposition and its relation to the polar decomposition

Higham, Nicholas J. (1994) The matrix sign decomposition and its relation to the polar decomposition. Linear Algebra and its Applications, 212-21. pp. 3-20. ISSN 0024-3795

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Abstract

The sign function of a square matrix was introduced by Roberts in 1971. We show that it is useful to regard S = sign(A) as being part of a matrix sign decomposition A = SN, where N = (A2)1/2. This decomposition leads to the new representation sign(A) = A(A2)−1/2. Most results for the matrix sign decomposition have a counterpart for the polar decomposition A = UH, and vice versa. To illustrate this, we derive best approximation properties of the factors U, H, and S, determine bounds for ||A − S|| and ||A − U||, and describe integral formulas for S and U. We also derive explicit expressions for the condition numbers of the factors S and N. An important equation expresses the sign of a block 2 × 2 matrix involving A in terms of the polar factor U of A. We apply this equation to a family of iterations for computing S by Pandey, Kenney, and Laub, to obtain a new family of iterations for computing U. The iterations have some attractive properties, including suitability for parallel computation.

Item Type: Article
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Ms Lucy van Russelt
Date Deposited: 05 Jul 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/361

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