7.1.8. Explain why the eigenvalues of A*A and AA* are real and nonneg- ative for every A Cmxn. Hint: Consider ||Ax|| / ||x||2. When are the eigenvalues of A*A and AA* strictly positive? Book solution: If (A, x) is an eigenpair for A*A, then ||Ax|| / ||x||² = x* A*Ax/x*x = \ is real and nonnegative. Furthermore, λ> 0 if and only if A*A is nonsingular or, equivalently, n = rank (A*A) = rank (A). Similar arguments apply to AA*. Please explain the book solution.

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