Real Matrix-Matrix Products

Definition

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 $B$.

Notation

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

Properties

Future editions will contain accounts of the following basic properties.

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. ↩︎