11.3 Matrix-Matrix Multiplication via Linear Combination of Rows

#Type/Definition #Topic/Linear_Algebra

Types: N/A
Examples: 11.3.1 Example of Matrix-Matrix Multiplication via Linear Combination of Rows
Constructions: N/A
Generalizations: 11.6 Algebraic Properties of Matrix-Matrix Multiplication

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Matrix-Matrix Multiplication via Linear Combination of Rows

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

{Row}_{i} (B) = {Row}_{i} (X) \cdot A

for $i \in {1, 2, \dots, m}$ .

The $i$ th row of $B$ is the matrix $A$ multiplied on the left by the $i$ th row of $X$ .

In 8.11 Colon Notation, we write this as

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

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

\begin{aligned} [\begin{array}{c} b_{i 1} & b_{i 2} & \dots & b_{i n} \end{array}] & = x_{i 1} [\begin{array}{c} a_{11} & a_{12} & \dots & a_{1 n} \end{array}] \\ + x_{i 2} [\begin{array}{c} a_{21} & a_{22} & \dots & a_{2 n} \end{array}] \\ ⋮ \\ + x_{i p} [\begin{array}{c} a_{p 1} & a_{p 2} & \dots & a_{p n} \end{array}] \end{aligned}

Note the similar structure to 10.4 Row-Vector-Matrix Multiplication via Linear Combinations.
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Remark. When doing matrix-matrix multiplication via linear combination of rows, we say that we multiply $A$ on the left by $X$ if $A$ is the right argument and $X$ is the left argument.

We want to manipulate the rows of matrix $A$ in some way
We think of $A$ as the modeling matrix and $X$ as the algebraic work (recall left matrix multiplication)

Left Matrix-Matrix Multiplication

$\underset{algebraic worker}{\overset{left factor}{X}} \cdot \underset{modeling matrix}{\overset{right factor}{A}} = \overset{product}{B}$

Start with modeling matrix $A$ in right argument
Strategically choose algebraic worker $X$
Hit $A$ on the left with $X$ (place $X$ in the left argument)
Produce the output product $B$

! The inner dimensions must agree!
- Number of rows of $X$ must equal number of columns of $A$
- If dimensions agree, $A$ is conformable to $X$ for left matrix-matrix multiplication

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