Let $D$ be a diagonal matrix and $A$ be a rank-$1$ matrix. I want to compute the matrix exponential $ e^{D+A}$, if possible by diagonalization of the matrix $D+A$. Is there a way of computing the diagonalization of this rank-$1$-correction of a diagonal matrix efficiently and in a numerically stable way, like the Sherman-Morrison-Woodbury formula for the inverse?
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Rodrigo de Azevedo
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Analysis801
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One approach is to use the algorithm explained in this post to get the eigenvalue decomposition of $D + A$, then apply the usual exponentiation algorithm for a diagonalizable matrix. – Ben Grossmann Nov 04 '21 at 15:46
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Unfortunately, this seems to require that the correction $A$ is symmetric, which is not the case for me. – Analysis801 Nov 04 '21 at 16:01
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1You're right, my mistake. Still, a similar approach might be applicable here. – Ben Grossmann Nov 04 '21 at 19:24
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Is $\bf D$ positive definite? – Rodrigo de Azevedo Jul 01 '23 at 11:29