The matrix sign decomposition and its relation to the polar decomposition (Q1344049)

The sign function maps a matrix \(A\) with complex entries to a matrix \[ S = \text{sign} (A) = Z \left( \begin{matrix} - I & 0 \\ - 0 & I \end{matrix} \right) Z^{-1}, \] if \(A = ZJZ^{-1}\), where \(J\) is the Jordan form of \(A\) and \(J = \left( \begin{smallmatrix} J_ 1 & 0 \\ 0 & J_ 2 \end{smallmatrix} \right)\), where the eigenvalues of \(J_ 1\) lie in the open left half plane and those of \(J_ 2\) lie in the open right half plane. The matrix sign decomposition of \(A\) is defined by \(A = SN\). The author shows that there are several relationships and analogies between the matrix sign decomposition and the polar decomposition of a matrix. He obtains the formula \(S = A(A^ 2)^{-1/2}\), derives some perturbation result, establishes error bounds and applies his results to a family of iterations for computing \(S\).