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Exercise 6.10.8

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Let $λ$ be an eigenvalue of $A A^{∗}$ . If $λ = 0$ , then we have $A A^{∗}$ is not invertible. Hence $A$ and $A^{∗}$ are not invertible, so is $A^{∗} A$ . So $λ$ is an eigenvalue of $A^{∗} A$ .

Suppose now that $λ \neq 0$ . We may find some eigenvector $x$ such that $A A^{∗} x = λx$ . This means that

A^{∗} A (A^{∗} x) = λ A^{∗} x .

Since $A^{∗} x$ is not zero, $λ$ is an eigenvalue of $A^{∗} A$ .

jephianlin

2011-06-27 00:00

Exercise 6.10.8

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