Cauchy Binet formula

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2010 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).


[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

F.R. Gantmacher, "The theory of matrices. Vol. 1", AMS Chelsea Publishing, Providence, RI, (1998). MR1657129

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