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. © 1994.
Original language | English |
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Pages (from-to) | 3-20 |
Number of pages | 17 |
Journal | Linear Algebra and its Applications |
Volume | 212-213 |
Issue number | C |
Publication status | Published - 15 Nov 1994 |