Last updated: March 10, 2023

When reading textbooks, papers, or work stuff, I occasionally find myself writing small proofs or derivations to better understand a particular topic. Eventually I’ll forget these derivations and have to look them up or re-derive them again.

This post is an attempt to gather such derivations in a central place for future use. I plan to occasionally update this post as new material comes up.



1.          \(\log \det(\mathbf{A}) = \mathrm{Tr}(\log \mathbf{A})\)

We show that \(\log \det(\mathbf{A}) = \mathrm{Tr}(\log \mathbf{A})\) for diagonalizable matrices. This property holds for general matricies, but we do not prove that here.

Suppose that \(\mathbf{A}\) is a diagonalizable matrix with eigenvalues \(\lambda_i\). Then,

\[\begin{align} \log \det(\mathbf{A}) &= \log \left( \prod_i \lambda_i \right) \\ &= \sum_i \log \lambda_i \\ &= \mathrm{Tr}(\log \mathbf{A}) \end{align}\]

The first line makes use of the identity \(\det(\mathbf{A}) = \prod_i \lambda_i\). The third line makes use of the identity \(\log(\mathbf{A}) = \mathbf{P}^{-1}(\log \boldsymbol{\Lambda})\mathbf{P}\) for diagonalizable \(\mathbf{A}\) and that \(\mathrm{Tr}(\mathbf{A}) = \sum_i \lambda_i\).