Matrix Multiplication is Homogeneous of Degree 1

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Theorem

Let $\mathbf A$ be an $m \times n$ matrix and $\mathbf B$ be an $n \times p$ matrix such that the columns of $\mathbf A$ and $\mathbf B$ are members of $\R^m$ and $\R^n$, respectively.

Let $\lambda \in \mathbb F \in \set {\R, \C}$ be a scalar.

Then:

$\mathbf A \paren {\lambda \mathbf B} = \lambda \paren {\mathbf A \mathbf B}$


Proof

Let $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{n p}$

\(\, \displaystyle \forall i \in \closedint 1 m, j \in \closedint 1 p: \, \) \(\displaystyle \mathbf A \paren {\lambda \mathbf B}\) \(=\) \(\displaystyle \lambda \sum_{k \mathop = 1}^n a_{i k} b_{k j}\) Definition of Matrix Product (Conventional) and Definition of Matrix Scalar Product
\(\displaystyle \) \(=\) \(\displaystyle \sum_{k \mathop = 1}^n a_{i k} \paren {\lambda b_{k j} }\)
\(\displaystyle \) \(=\) \(\displaystyle \mathbf A \paren {\lambda \mathbf B}\) Definition of Matrix Product (Conventional) and Definition of Matrix Scalar Product

$\blacksquare$