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Cauchy-Binet as a generalized Pythagoras theorem.

Let $X$ be an $n \times k$ matrix with $n \ge k$. For any $k$-index $I=i_1...i_k, \; 1 \le i_1 <... < ... < i_k \le n$, there is some advantage to denote by $X_I$, the determinant of the $k \times k$ submatrix of $X$ with rows indexed by $I$. For any two such $X,Y$, we can state the Cauchy-Binet formula as a pairing $$\det (X^TY)= \sum_{I} X_I Y_I$$ where the sum is over all $n \choose k$ $k$-indices. This is a Pythagoras theorem for $X=Y$ since it says that the the volume-squared of the parallelepiped spanned by the $k$ columns of $X$ in $\mathbb{R}^n$ is the sum of squares of the volume of the projections on the $n \choose k$ $k$-dimensional coordinates.

For any $n \times m$ matrix $A$ with $m,n \ge k$ and $k$ indices $I,J$, we also denote by $A_{IJ}$ the determinant of the $k \times k$ submatrix of $A$ with rows indexed by $I$ and column indexed by $J$. Then for $X(m \times k)$ and $Y(n \times k)$, we have by Cauchy-Binet twice, $$\det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$ where $A^I$ is the $k \times n$ matrix given by the rows of $A$ indexed by $I$ and we note that $(AY)_I= \det(A^IY)$ and $(A^I)^T_J=A_{IJ}$. This notation thus allows us to view Cauchy-Binet (usually stated with $m=n,A=I$) as an extension of the usual $x^TAy=\sum_{ij}A_{ij}x_iy_j$ for $k=1$.

Let $X$ be an $n \times k$ matrix with $n \ge k$. For any $k$-index $I=i_1...i_k, \; 1 \le i_1<... For any$n \times m$matrix$A$with$m,n \ge k$and$k$indices$I,J$, we also denote by$A_{IJ}$the determinant of the$k \times k$submatrix of$A$with rows indexed by$I$and column indexed by$J$. Then for$X(m \times k)$and$Y(n \times k)$, we have by Cauchy-Binet twice, $$\det(X^TAY)=\det(X^T(AY))=\sum_{I}X_I(AY)_I =\sum_I X_I \det(A^IY)=\sum_I X_I \sum_J A_{IJ} Y_J,$$ where$A^I$is the$k \times n$matrix given by the rows of$A$indexed by$I$and we note that$(AY)_I= \det(A^IY)$and$(A^I)^T_J=A_{IJ}$. This notation thus allows us to view Cauchy-Binet (usually stated with$m=n,A=I$) as an extension of the usual$x^TAy=\sum_{ij}A_{ij}x_iy_j$for$k=1\$.