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$\newcommand\R{\mathbb{R}}\newcommand\C{\mathbb{C}}\newcommand\Z{\mathbb{Z}}$ Chain Rule for Map between Open Subsets of Euclidean Space Let $M$ be an open subset of $\R^m$, $N$ be an open subset of $\R^n$, and $O$ be an open subset of $\R^p$. Let …

$\newcommand\F{\mathbb{F}}\newcommand\R{\mathbb{R}}\newcommand\C{\mathbb{C}}\newcommand\Z{\mathbb{Z}}\newcommand\tr{\operatorname{trace}}\newcommand\End{\operatorname{End}}$ The trace of a sqaure matrix $A$ is defined to be the sum of the elements along the diagonal. A basic fact is that for any invertible matrix $M$, \begin{equation}\tag{*} \tr(M^{-1}AM) = \tr(A). \end{equation} …

$\newcommand\R{\mathbb{R}}\newcommand\C{\mathbb{C}}\newcommand\Z{\mathbb{Z}}$ The singular value decomposition is usually defined for a matrix. Here, we will show directly that any linear map between inner product spaces has a singular value decomposition. The singular value decomposition of a …