1.2
Matrix Algebra
Scalar Multiplication
Given a matrix \(A\) of size \( (m\times n) \), a scalar k can scale a matrix. This is accomplished by scaling each element by the scalar k.
\[k\cdot A=[ka_{ij}]\]- \( 3\begin{bmatrix} 1 & 0 & -1 \\ 2 & 4 & 6 \\ \end{bmatrix}\) \[\begin{align} A&=\begin{bmatrix} 3(1) & 3(0) & 3(-1) \\ 3(2) & 3(4) & 3(6) \\ \end{bmatrix} \\ &=\begin{bmatrix} 3 & 0 & -3 \\ 6 & 12 & 18 \\ \end{bmatrix} \end{align}\]
Addition
Given \(A\) and \(B\), matrices of the same size \( (m\times n) \), addition is given by:
\[ A+B=[a_{ij}+b_{ij}] \]- \( \begin{bmatrix} 2 & 0 \\ -1 & 4 \\ \end{bmatrix} + \begin{bmatrix} 6 & 5 \\ 3 & 7 \\ \end{bmatrix}\) \[A=\begin{bmatrix} 8 & 5 \\ 2 & 11 \\ \end{bmatrix}\]
Matrix Algebra Rules
Just like traditional algebra, matrix algebra has special rules defining how and when computations are performed.
- Commutative property: \(A+B=B+A\)
- Associative property: \(A+(B+C)=(A+B)+C\)
- Unit identity: \(A\cdot 1=A\)
- Distributive properties: \(s(A+B)=sA+sB\) and \((s+t)(A)=sA+tA\)
- Additional associative property: \((st)(A)=s(tA)\)
Subtraction
Given \(A + B\), both \((m\times n)\) matrices of the exact same size, subtraction is given by:
\[ A-B=A+(-1\cdot B) \]A zero matrix is given by:
\[\begin{bmatrix} A-A=0_{m\times n} \end{bmatrix}\]If no subscript is provided to a zero matrix identifier, the zero matrix is adjusted to be whatever size is needed.
\[A+0=A\]Vector Multiplication
Vector multiplication, or a vector dot product, is given formally by:
\[\begin{align} \overrightarrow v \cdot \overrightarrow w &= \langle v_1,v_2,v_3\rangle \cdot \langle w_1,w_2,w_3 \rangle \\ &=v_1w_1 + v_2w_2 + v_3w_3 \\ &=|\overrightarrow v||\overrightarrow w|\cos\theta \end{align}\]In a matrix context, this will be a row dot-producted with a column.