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2006.178: The matrix sign decomposition and its relation to the polar decomposition

2006.178: Nicholas J. Higham (1994) The matrix sign decomposition and its relation to the polar decomposition. Linear Algebra and its Applications, 212-213. pp. 3-20. ISSN 0024-3795

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DOI: 10.1016/0024-3795(94)90393-X

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 2000 > 15 Linear and multilinear algebra; matrix theory
MSC 2000 > 65 Numerical analysis
MIMS number:2006.178
Deposited By:Miss Louise Stait
Deposited On:05 July 2006

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