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I have a question that mentioned in the book "Matrix Computations" by Golub and van Loan. "Show that if $A\in \mathbb{R}^{n\times n}$ is an upper bidiagonal matrix having a repeated singular value, then $A$ must have a zero on its diagonal or superdiagonal."

I have proved this question is right for an upper bidiagonal matrix $A\in \mathbb{R}^{2\times 2}$. But I can not prove it for general upper bidiagonal matrices $A\in \mathbb{R}^{n\times n}$.

David Mayar
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For sake of having an answer, note that if all diagonal and superdiagonal elements of $A$ are nonzero, then $A^TA$ would be a symmetric tridiagonal matrix with nonzero super/sub-diagonal entries. It is known that such a matrix must have distinct eigenvalues (see the proof by Algebraic Pavel or the answer by Chris Godsil in two other threads). It follows that $A$ has distinct singular values.

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