(b) If *A* is singular, there is no *X* ϵ *M*_{n} such that e^{X} = *A*.

(c) Suppose *A* is real. There is a real *X* ϵ *M*_{n}() such that e^{X} = *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 e^{X} = *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 *M*_{n} be given.

(a) If *A* is nonsingular, then *A = GQ*, where *Q* ϵ *M*_{n} is complex orthogonal (*QQ*^{T} = *I*), *G* ϵ

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