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I would like to calculate the Adjugate matrix of a given matrix $A$, and its updates in the diagonal: $B=A-\lambda I$, where $I$ is the identity matrix, $\lambda$ is a scalar. To this end, I am using the algorithm explained here, using a decomposition $A=XDY$. Unfortunately, the matrix $A$ or $B$ could be singular.

Is it possible to obtain a suitable decomposition such, that for the decomposition of $B$ I could save computer time by reusing $X$, $D$, or $Y$ from the decomposition of $A$?

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  • $\begingroup$ One attempt that I have made is that I perform SVD on A'=A+I in order to remove the null-space, and decompose it as A'=X'D'Y'. Then I calculate D"=abs(D'-I) as my matrices are positive semi-definite, symmetric and real. Numerical tests in python are provided spot-on results for the adjugate using X', D" (or D"-lambda*I), and Y' but I would like to see the limits of this approach. Any suggestions? $\endgroup$ Commented Nov 23, 2020 at 15:46

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