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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.

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.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 (x,x) is an eigenpair for A*A, then ||Ax||²2 / ||x||2²2 = x*A*Ax/x*x = \ is real and nonnegative. Furthermore, > > 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.
Transcribed Image Text: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 (x,x) is an eigenpair for A*A, then ||Ax||²2 / ||x||2²2 = x*A*Ax/x*x = \ is real and nonnegative. Furthermore, > > 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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