English
Let A,B be 3×3 matrices. Their product AB is the 3×3 matrix with entries given by the standard triple sum: (AB)_{ij} = sum over k of A_{ik} B_{kj}.
Русский
Пусть A,B — матрицы 3×3. Произведение AB — матрица 3×3, элементы которой задаются суммой по k: (AB)_{ij} = ∑_{k=1}^3 A_{ik} B_{kj}.
LaTeX
$$$\begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{pmatrix} = \begin{pmatrix} a_{11} b_{11} + a_{12} b_{21} + a_{13} b_{31} & a_{11} b_{12} + a_{12} b_{22} + a_{13} b_{32} & a_{11} b_{13} + a_{12} b_{23} + a_{13} b_{33} \\ a_{21} b_{11} + a_{22} b_{21} + a_{23} b_{31} & a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32} & a_{21} b_{13} + a_{22} b_{23} + a_{23} b_{33} \\ a_{31} b_{11} + a_{32} b_{21} + a_{33} b_{31} & a_{31} b_{12} + a_{32} b_{22} + a_{33} b_{32} & a_{31} b_{13} + a_{32} b_{23} + a_{33} b_{33} \end{pmatrix}$$$
Lean4
/-- The permanent of a square matrix defined as a sum over all permutations. This is analogous to
the determinant but without alternating signs. -/
def permanent (M : Matrix n n R) : R :=
∑ σ : Perm n, ∏ i, M (σ i) i
