11.2 Matrix-Matrix Multiplication via Linear Combination of Columns

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 11.2.1 Example of Matrix-Matrix Multiplication via Linear Combination of Columns
Constructions: N/A
Generalizations: 10.1 Matrix-Column-Vector Multiplication via Linear Combinations

Properties: 11.6 Algebraic Properties of Matrix-Matrix Multiplication
Sufficiencies: N/A
Questions: N/A

Matrix-Matrix Multiplication via Linear Combination of Columns

Let matrices $A \in R^{m \times p}$ and $X \in R^{p \times n}$ . If we multiply $A$ on the right by $X$ to form the $m \times n$ matrix $B = A \cdot X$ , then

{Column}_{k} (B) = A \cdot {Column}_{k} (X)

for $k \in {1, 2, \dots, n}$ .

The $k$ th column of $B$ is the matrix $A$ multiplied on the right by the $k$ th column of $X$ .

In 8.11 Colon Notation, we write this operation as

B (:, k) = A \cdot X (:, k)

Remark. The $k$ th column of product output $B$ is a linear combination of the columns of matrix $A$ and the scalars of the $k$ th column of $X$ . We can write this as

B (:, k) = {[\begin{matrix} b_{1 k} \\ b_{2 k} \\ ⋮ \\ b_{m k} \end{matrix}]}_{m \times 1} = x_{1 k} {[\begin{matrix} a_{11} \\ a_{21} \\ ⋮ \\ a_{m 1} \end{matrix}]}_{m \times 1} + x_{2 k} {[\begin{matrix} a_{12} \\ a_{22} \\ ⋮ \\ a_{m 2} \end{matrix}]}_{m \times 1} + \dots + x_{p k} {[\begin{matrix} a_{1 p} \\ a_{2 p} \\ ⋮ \\ a_{m p} \end{matrix}]}_{m \times 1}

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