Question

Prove that if the square matrix $A$ is nonsingular, then the singular values of $A^{-1}$ are the reciprocals of the singular values of $A$. How are their condition numbers related?

    Prove that if the square matrix $A$ is nonsingular, then the singular values of $A^{-1}$ are the reciprocals of the singular values of $A$. How are their condition numbers related?

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 8, Problem 12

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If $A$ is an $n \times n$ matrix, its singular values are the square roots of the eigenvalues of $A^*A$, where $A^*$ denotes the conjugate transpose of $A$. Let $\sigma_1, \sigma_2, \ldots, \sigma_n$ be the singular values of $A$. Show more…

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Prove that if the square matrix $A$ is nonsingular, then the singular values of $A^{-1}$ are the reciprocals of the singular values of $A$. How are their condition numbers related?

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