User:Matteo.focardi/sandbox
2020 Mathematics Subject Classification: Primary: 15Axx [MSN][ZBL]
A formula aimed at expressing the determinant of the product of two matrices 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's 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's 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} Cauchy-Binet's 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 Cauchy-Schwartz's 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's 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}^n\to\mathbb{R}^m is the linear map associated to A^T, and Q\subset\mathbb{R}^n is the unitary cube, the n-th dimensional volume of the parallelepiped \mathcal{A}(Q)\subset\mathbb{R}^m is given by \sqrt{\det(A^T\,A)} due to polar decomposition of A, recall that n\leq m.
Formula (1) above then expresses the square of the n-th dimensional volume of \mathcal{A}(Q) as the sum of the squares of the volumes of the projections on all coordinates n 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 |
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28852