Rademacher theorem

2020 Mathematics Subject Classification: Primary: 15Axx [MSN][ZBL]

A formula aimed at expressing the determinant of the product of two matrices $A\in\mathrm{M}_{m,n}(\mathbb{R})$ and $B\in\mathrm{M}_{n,m}(\mathbb{R})$, in terms of the sum of the products of all possible higher order minors of $A$ with corresponding minors of the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$ denotes any multi-index $(\beta_1,\ldots,\beta_m)$ with $1\leq \beta_1<\ldots<\beta_m\leq n$ of length $m$, then \[ \det(AB)=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\alpha}, \] where $A_{\alpha\,\beta}=(a_{\alpha_i\beta_j})$ and $B_{\beta\,\alpha}=(a_{\beta_j\alpha_i})$. In case $m>n$, no such $\beta$ exists and the right-hand side above is set to be $0$ by definition.

Note that if $n=m$ the formula reduces to \[ \det (AB)=\det A\,\det B. \] More generally, if $A\in\mathrm{M}_{m,n}(\mathbb{R})$, $B\in\mathrm{M}_{n,q}(\mathbb{R})$ and $p\leq\min\{m,q\}$, then any minor of order $p$ of the product matrix $AB$ can be expressed as follows by Cauchy Binet formula \[ \det((AB)_{\alpha\,\gamma})=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\gamma}, \] where $\alpha=(\alpha_1\ldots,\alpha_p)$ with $1\leq\alpha_1<\ldots<\alpha_p\leq m$, $\gamma=(\gamma_1,\ldots,\gamma_p)$ with $1\leq\gamma_1<\ldots<\gamma_p\leq q$, and $\beta=(\beta_1,\ldots,\beta_m)$ with $1\leq \beta_1<\ldots<\beta_m\leq n$.

A number of interesting consequence of Cauchy Binet formula is listed below. First of all, an inequality for the rank of the product matrix follows straightforwardly, i.e., \[ \mathrm{rank}(AB)\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}. \] Moreover, if $m=2$, $\mathbf{a}$, $\mathbf{b}\in\mathbb{R}^n$ are two vectors, by taking $$A=\begin{pmatrix} a_{1}&\dots&a_{n}\\ b_{1}&\dots&b_{n}\\ \end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix} a_{1}&b_{1}\\ \dots&\dots\\ a_{n}&b_{n}\\ \end{pmatrix} $$ the Cauchy Binet formula yields \[ \sum_{1\leq i<j\leq n}\begin{vmatrix} a_{i}&a_{j}\\ b_{i}&b_{j}\\ \end{vmatrix}^2= \begin{vmatrix} \|\mathbf{a}\|^2&\langle\mathbf{a},\mathbf{b}\rangle\\ \langle\mathbf{a}, \mathbf{b}\rangle&\|\mathbf{b}\|^2\\ \end{vmatrix}, \] in turn implying the Cauchy Schwarz inequality. Here, $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$ are the Euclidean norm and scalar product, respectively.

Let us finally interpret geometrically the result. Take $B=A^T$, then $\det(A_{\alpha\beta})=\det(A^T_{\beta,\alpha})$, so that by Cauchy Binet formula \[\label{p} \det(A^T\,A)=\sum_\beta(\det(A_{\alpha\beta}))^2. \] This is a generalization of the Pythagorean formula, corresponding to $m=1$. Indeed, if $\mathcal{B}:\mathbb{R}^m\to\mathbb{R}^n$ is the linear map associated to $A^T$, and $Q\subset\mathbb{R}^m$ is the unitary cube, the $m$-th dimensional volume of the parallelepiped $\mathcal{A}(Q)\subset\mathbb{R}^n$ is given by $\sqrt{\det(A^T\,A)}$ due to the polar decomposition of $A$, recall that $m\leq n$.

Formula (1) above then expresses the square of the $m$-th dimensional volume of $\mathcal{A}(Q)$ as the sum of the squares of the volumes of the projections on all coordinates $m$ planes (cp. with Area formula).

References

[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]	F.R. Gantmacher, "The theory of matrices. Vol. 1", AMS Chelsea Publishing, Providence, RI, (1998). MR1657129