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I have recently learned about the representation theorem for isotropic, linear operators, which says the following: Defintion: Let $M_n$ be the vector space of $n \times n$ real matrices. We say a l …

$\newcommand{\Sig}{\Sigma}$ $\newcommand{\dist}{\operatorname{dist}}$ $\newcommand{\distSO}[1]{\dist(#1,\SO)}$ $\newcommand{\distO}[1]{\text{dist}(#1,\On)}$ $\newcommand{\tildistSO}[1]{\operatorname{d …

$\newcommand{\GLm}{\text{GL}_n^-}$Let $A$ be a real $n \times n$ matrix with non-positive determinant. Suppose that the smallest singular value of $A$ is strictly smaller than all the others (it has m …

This is somewhat informal. My argument is in the spirit of dimensional analysis in physics. The essential point is that the cofactor matrix encapsulates $n-1$ dimensional volume, while the dimension …

This is a more detailed version of fedja's answer: We shall need the following preliminary results: Lemma 1: Let $\lambda \in \mathbb{S}^1$. Then, the distance of $\lambda$ from $1$ is not greater t …

$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\t …

Every $A \in \text{GL}_n(\mathbb{R})$ has a unique orthogonal polar factor $O_A=A(\sqrt{A^TA})^{-1}$, ( $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$see Polar decomposition). …

Every $A \in \text{GL}_n(\mathbb{R})$ has a unique Polar decomposition: $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$. In particular the orthogonal factor is given by $$O_A=A(\ …

Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The cofactor matrix of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$. Thus, it is immediate that $A \in \ope …