The Matrix Sign Decomposition and Its Relation to the Polar Decomposition

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 = (A ) ’ 1/z This decomposition leads to . the new representation sign(A) = A(A2)-‘i2. 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 I/A SII and IIA UII, 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 x 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.