Matrix Scalar Product is Associative
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Theorem
Let $\Bbb F$ denote one of the standard number systems.
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.
For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$.
The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense:
For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$:
- $\lambda \paren {\mu \mathbf A} = \paren {\lambda \mu} \mathbf A$
Proof
\(\ds \lambda \paren {\mu \mathbf A}\) | \(=\) | \(\ds \lambda \paren {\mu \sqbrk a_{m n} }\) | Definition of $\mathbf A$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lambda \paren {\sqbrk {\mu a}_{m n} }\) | Definition of Matrix Scalar Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {\lambda \paren {\mu a} }_{m n}\) | Definition of Matrix Scalar Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {\paren {\lambda \mu} a}_{m n}\) | Associative Law of Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lambda \mu} \sqbrk a_{m n}\) | Definition of Matrix Scalar Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\lambda \mu} \mathbf A\) | Definition of $\mathbf A$ |
$\blacksquare$
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices: $4$