Let A, B be two symmetric positive-definite matrices. Let $\lambda_i$ be the generalized eigenvalues of the pencil (A,B). Can we write function $\log\frac{|A|}{|B|}$ (where $|\cdot|$ stands for determinant) in terms of $\lambda_i$? thanks!
2026-04-02 10:16:48.1775125008
Determinant and generalized eigenvalues
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hint: Let $B=R^2$. Then $$\det{(R^{-1}AR^{-1})}=\det{(AR^{-2})}=\det{(A)}\det{(R^{-2})}=\frac{\det{(A)}}{\det{(B)}}$$ (verify!!)
How are generalized eigenvalues of the pencil $(A,B)$ related to the eigenvalues of $(R^{-1}AR^{-1})$?