Given non-singular real matrix $A$ of size $n \times n$. I want to find a non-singular diagonal matrix $D$ such that $$ B = D A D^{-1} $$ has singular values $\sigma(B) = [1, \dots, 1, \lvert \det(A) \rvert]$. First, how to determine that such $D$ exists? Second, how to (numerically) compute it?
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$\begingroup$ This is hopeless. If $A$ is diagonal, then $B=A$. (There are also other problems for more general $A$.) $\endgroup$– Christian RemlingCommented Dec 23, 2020 at 16:11
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$\begingroup$ @ChristianRemling I agree that there are many cases in which there is no solution, but I’m expecting more a numerical test to determine that a solution exists. $\endgroup$– JiroCommented Dec 23, 2020 at 16:55
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