(b) If A is singular, there is no X ϵ Mn such that eX = A.
(c) Suppose A is real. There is a real X ϵ Mn() such that eX = A if and only if A is nonsingular and has an even number of Jordan blocks of each size for every negative eigenvalue. If A has any negative eigenvalues, no real solution of eX = A can be a polynomial in A or a primary matrix function of A.
One interesting application of the square root primary matrix function is to prove the following analog of the classical polar decomposition.
6.4.16 Theorem. Let A ϵ Mn be given.
(a) If A is nonsingular, then A = GQ, where Q ϵ Mn is complex orthogonal (QQT = I), G ϵ