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Sparse dense matrix versus non-sparse dense matrix in eigenvalue computation

If you take the transpose of the matrix, which has the same eigenvalues, and you divide by $w_0$, you get the (block) companion matrix of a quadratic matrix polynomial with $n \times n$ coefficients. ...
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Solving linear matrix equation

I feel like you are drowning in a glass of water. Putting $AC=U\in M_2$ and $BC=V\in M_2$, we obtain (*) $UXU^T-VXV^T=L$ in $M_2$. If $X$ is a symmetric solution of (*), then $L$ too. For generic $U,V,...
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1 vote
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On the eigen vectors of a diagonalizable matrix

I mentioned the silly example $U = A = D$ offhand, but, in fact, it's essentially the only example. In general, note that the columns of $U^{-1}$ are the eigenvectors of $A$. So we're asking for, at ...
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4 votes
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Total positivity tests: optimal in the number of minors vs. the computational cost

There are at least a few places looking at complexity for total positivity counting arithmetic operations. I don't know if people have looked at trying to find smaller minors. These use initial minors ...
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