Subsection VSP Vector Space Properties
With definitions of matrix addition and scalar multiplication we can now state, and prove, several properties of each operation, and some properties that involve their interplay. We now collect ten of them here for later reference.
Theorem VSPM Vector Space Properties of Matrices
Suppose that $M_{mn}$ is the set of all $m\times n$ matrices (Definition VSM) with addition and scalar multiplication as defined in Definition MA and Definition MSM. Then
- ACM Additive Closure, Matrices
If $A,\,B\in M_{mn}$, then $A+B\in M_{mn}$.
- SCM Scalar Closure, Matrices
If $\alpha\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha A\in M_{mn}$.
- CM Commutativity, Matrices
If $A,\,B\in M_{mn}$, then $A+B=B+A$.
- AAM Additive Associativity, Matrices
If $A,\,B,\,C\in M_{mn}$, then $A+\left(B+C\right)=\left(A+B\right)+C$.
- ZM Zero Matrix, Matrices
There is a matrix, $\zeromatrix$, called the zero matrix, such that $A+\zeromatrix=A$ for all $A\in M_{mn}$.
- AIM Additive Inverses, Matrices
If $A\in M_{mn}$, then there exists a matrix $-A\in M_{mn}$ so that $A+(-A)=\zeromatrix$.
- SMAM Scalar Multiplication Associativity, Matrices
If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha(\beta A)=(\alpha\beta)A$.
- DMAM Distributivity across Matrix Addition, Matrices
If $\alpha\in\complex{\null}$ and $A,\,B\in M_{mn}$, then $\alpha(A+B)=\alpha A+\alpha B$.
- DSAM Distributivity across Scalar Addition, Matrices
If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then
$(\alpha+\beta)A=\alpha A+\beta A$.
- OM One, Matrices
If $A\in M_{mn}$, then $1A=A$.
For now, note the similarities between Theorem VSPM about matrices and Theorem VSPCV about vectors.
The zero matrix described in this theorem, $\zeromatrix$, is what you would expect — a matrix full of zeros.
Definition ZM Zero Matrix
The $m\times n$ zero matrix is written as $\zeromatrix=\zeromatrix_{m\times n}$ and defined by $\matrixentry{\zeromatrix}{ij}=0$, for all $1\leq i\leq m$, $1\leq j\leq n$.