\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Real Matrix-Vector Products
Needed by:
Matrix Determinant of Product
Matrix Rings
Matrix Squares
Matrix Trace
Quadratic Forms
Real Matrices with Orthonormal Columns
Real Matrix Inverses
Rooted Tree Linear Cascades
Sheet PDF
Graph PDF

Real Matrix-Matrix Products


Let $A \in \R ^{l \times m}$ and $B \in \R ^{m \times n}$. In this case we call $A$ and $B$ conformable. The matrix-matrix product of $A$ and $B$ is the matrix $C \in \R ^{l \times n}$ whose $i$th row $c_i$ (for $i = 1, \dots , n$) is defind $c_i = Ab_i$ where $b_i$ is the $i$th row of $B$.


We denote the matrix product of $A$ and $B$ by $AB$.


Future editions will contain accounts of the following basic properties.

Matrix multiplication is associative.
Matrix multiplication is not commutative.

Indeed, the matrix-matrix product of $B$ and $A$ may not even be defined, if $B$ and $A$ are not conformable.

Identity matrix

The matrix which is the identity under the operation of multiplication is the one which has ones on its diagonals and zero elswhere.1 We denote the $d \times d$ identity matrix by $I_d$, or, when no confusion is possible by $I$.

  1. Future editions will improve and expand. ↩︎
Copyright © 2023 The Bourbaki Authors — All rights reserved — Version 13a6779cc About Show the old page view