Every Invertible Real Matrix Is an Orthogonal Matrix Times a Positive Diagonal Matrix Times an Orthogonal Matrix

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $n\in\mathbb{N}$ and let $A\in\mathcal{M}_{n}(\mathbb{R})$ be invertible.

(i) Singular value decomposition. There are $U,V\in\mathrm{O}(n)$ and $d\in\mathbb{R}^{n}$ with $d_{k}>0$ for every $k\in[n]$ such that

$$A=U\operatorname{diag}(d)V^{\top}.$$

(ii) The determinant. Let $U,V\in\mathrm{O}(n)$ and $d\in\mathbb{R}^{n}$ with $d_{k}>0$ for every $k\in[n]$ satisfy $A=U\operatorname{diag}(d)V^{\top}$. Then $|\det U|=|\det V|=1$ and

$$|\det A|=\det\operatorname{diag}(d)=\prod_{k=1}^{n}d_{k}.$$