MATH 240

Fri. September 20th, 2019

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Matrix Operations

Addition

Matrices of the same dimentions can be added together element-wise.

Scaling

Multiplying a matrix by a scalar multiplies every element of the matrix by that scalar.

Multiplication

For two matrices $A,B$, for $AB$ to be defined, the number of columns in $A$ must be the same as the number of rows in $B$.

If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ will be $m\times p$.

If you lay the $i$th row of $A$ over the $j$th column of $B$, multiply element-wise, and add the results together, you get element $i,j$ of $AB$.

Ex.
$\begin{aligned} A&=\begin{pmatrix} 2&3\\1&-5 \end{pmatrix}\\ B&=\begin{pmatrix} 4&3&6\\1&-2&3 \end{pmatrix}\\ \\ AB&=\begin{pmatrix} 2(4)+3(1)&2(3)+3(-2)&2(6)+3(3)\\ 1(4)+(-5)(1)&1(3)+(-5)(-2)&1(6)+(-5)(3) \end{pmatrix}\\ &=\begin{pmatrix} 11&0&21\\-1&13&-9 \end{pmatrix} \end{aligned}$

Transpose

For a matrix $A$, its transpose $A^T$ is the matrix whose columns are the rows of $A$.

Ex.
$\begin{aligned} A&=\begin{pmatrix} a&b\\c&d \end{pmatrix}\\ A^T&=\begin{pmatrix} a&c\\b&d \end{pmatrix} \end{aligned}$

Properties of Matrix Operations

If $A,B,C$ are matrices, $r,s$ are scalars, and $I_m$ is an identity matrix:
$\begin{aligned} A+B&=B+A\\ (A+B)+C&=A+(B+C)\\ A+0&=A\\ r(A+B)&=rA+rB\\ (r+s)A&=rA+sA\\ (rs)A&=r(sA)\\ \\ A(BC)&=(AB)C\\ A(B+C)&=AB+AC\\ (B+C)A&=BA+CA\\ r(AB)&=(rA)B=A(rB)\\ I_mA&=A=AI_m\\ \\ (A^T)^T&=A\\ (A+B)^T&=A^T+B^T\\ (AB)^T&=B^TA^T. \end{aligned}$

Note that $AB=BA$ is not necessarily true! (Matrix multiplication is not commutative.)

Identity Matrices

An identity matrix $I_m$ is a $m\times m$ matrix with $1$s along the diagonal and $0$s everywhere else.

Ex.
$I_4=\begin{pmatrix} 1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1 \end{pmatrix}$

This acts as an analogue to $1$ for matrices – it is the multiplicative identity.