Editing Cauchy–Binet formula (section)

== Continuous version ==

A continuous version of the Cauchy–Binet formula, known as the '''Andréief identity''',<ref>C. Andréief, "Note sur une relation entre les intégrales définies des produits des fonctions", ''Mémoires de la Société des Sciences Physiques et Naturelles de Bordeaux'' (3) 2 (1886), 1–14</ref> appears commonly in random matrix theory.<ref>{{cite book|last=Mehta|first=M.L.|title=Random Matrices|publisher=Elsevier/Academic Press|location=Amsterdam|year=2004|edition=3rd|isbn=0-12-088409-7}}</ref> It is stated as follows: let <math>\left\{f_j(x)\right\}_{j=1}^{N}</math> and <math>\left\{g_j(x)\right\}_{j=1}^{N}</math> be two sequences of integrable functions, supported on <math>I</math>. Then
:<math>\int_I \cdots \int_I \det \left[f_{j}(x_k)\right]_{j,k=1}^N \det \left[g_{j}(x_k)\right]_{j,k=1}^N dx_1 \cdots dx_N 
= N!\, \det \left[\int_I f_j(x)g_k(x) dx\right]_{j,k=1}^{N}.</math>

{{Math proof|title=Proof|proof=
 Let <math> S_N</math> be the [[permutation group]] of order N, <math>|s|</math> be the sign of a permutation, <math>\langle f, g \rangle = \int_I f(x) g(x) dx </math> be the "inner product".<math display="block">\begin{align}
\text{left side} &= \sum_{s, s' \in S_N} (-1)^{|s| + |s'|} \int_{I^N} \prod_{j} f_{s(j)}(x_j) \prod_k g_{s'(k)}(x_k)\\
 &=\sum_{s, s' \in S_N} (-1)^{|s| + |s'|} \int_{I^N} \prod_{j} f_{s(j)}(x_j) g_{s'(j)}(x_j)\\
 &=\sum_{s, s' \in S_N} (-1)^{|s| + |s'|}  \prod_{j} \int_{I} f_{s(j)}(x_j) g_{s'(j)}(x_j) d x_j\\
 &=\sum_{s, s' \in S_N} (-1)^{|s| + |s'|}   \prod_{j}\langle f_{s(j)}, g_{s'(j)} \rangle\\
 &=\sum_{s' \in S_N} (-1)^{|s'| + |s'|}  \sum_{s \in S_N} (-1)^{|s| + |s'^{-1}|} \prod_{j} \langle f_{(s\circ s'^{-1})(j)}, g_{j} \rangle\\
 &=\sum_{s' \in S_N}\sum_{s \in S_N} (-1)^{|s \circ s'^{-1}|}  \prod_{j} \langle f_{(s\circ s'^{-1})(j)}, g_{j} \rangle\\
 &= \text{right side}\\
\end{align}</math> 
}}

Forrester<ref>{{cite arXiv |eprint=1806.10411 |title=Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882–83 |last=Forrester|first=Peter J. |date=2018|class=math-ph }}</ref> describes how to recover the usual Cauchy–Binet formula as a discretisation of the above identity.

{{Math proof|title=Proof|proof=
Pick <math>t_1 < \cdots < t_m</math> in <math>[0, 1]</math>, pick <math>f_1, \ldots, g_n</math>, such that <math>f_j(t_k) = A_{j, k}</math> and the same holds for <math>g</math> and <math>B</math>. Now plugging in <math>f_j(x_k) = \sum_l A_{j,l} \delta(x_k - t_l)</math> and <math>g_j(x_k) = \sum_l B_{j,l} \delta(x_k - t_l) </math> into the Andreev identity, and simplifying both sides, we get:
<math display="block">\sum_{l_1, \ldots, l_n \in [1:m]} \det [f_j(t_{l_k})] \det [g_j(t_{l_k})] = n! \det \left[\sum_l f_j(t_{l}) g_k(t_l)\right]</math>
The right side is <math>n! \det(AB)</math>, and the left side is <math>n! \sum_{S\subset [1:m], |S| = n} \det(A_{[1:m],S})\det(B_{S,[1:m]})</math>.
}}It is occasionally called the '''[[Konstantin Andreev|Andréief]]-[[Eduard Heine|Heine]] identity''', though the credit to Heine sees unhistorical, as pre-2010 sources generally credit only Andréief.<ref>{{Cite web |title=What's a Heine reference for the "Andréief-Heine identity" |url=https://hsm.stackexchange.com/questions/17734/whats-a-heine-reference-for-the-andr%c3%a9ief-heine-identity/17735#17735 |access-date=2025-04-20 |website=History of Science and Mathematics Stack Exchange |language=en}}</ref>