4.3b Matrix-Matrix Multiplication

Matrix-Matrix Multiplication

Two matrices $A\in {\mathbb{R}}^{m\times n}$ and $B\in {\mathbb{R}}^{n\times p}$ can be multiplied to create a matrix $A\times B\in {\mathbb{R}}^{m\times p}$, such that

$$AB=\left[\begin{array}{c}----{\vec{a}}_1^T----\\ ----{\vec{a}}_2^T----\\ ⋮\\ ----{\vec{a}}_m^T----\end{array}\right]\left[\begin{array}{cccc}|& |& & |\\ |& |& & |\\ |& |& & |\\ {\vec{b}}_1& {\vec{b}}_2& \cdots & {\vec{b}}_p\\ |& |& & |\\ |& |& & |\\ |& |& & |\end{array}\right]=\left[\begin{array}{cccc}{\vec{a}}_1^T{\vec{b}}_1& {\vec{a}}_1^T{\vec{b}}_2& \dots & {\vec{a}}_1^T{\vec{b}}_p\\ {\vec{a}}_2^T{\vec{b}}_1& {\vec{a}}_2^T{\vec{b}}_2& \dots & {\vec{a}}_2^T{\vec{b}}_p\\ ⋮& ⋮& & ⋮\\ {\vec{a}}_m^T{\vec{b}}_1& {\vec{a}}_m^T{\vec{b}}_2& \dots & {\vec{a}}_m^T{\vec{b}}_p\end{array}\right],$$

where each element of the resulting matrix $D_{ij}={\vec{a}}_i^T{\vec{b}}_j$ equals the inner product of the left ($A$) matrix’s $i$th row vector and the right ($B$) matrix’s $j$th column vector.

Non-Commutativity $AB\ne BA$.

The matrix product is non-commutative, meaning that, in general,

Properties