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Topics in Matrix Analysis by Charles R. Johnson, Roger A. Horn

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  (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(image) 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

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