Matrix Addition, Scalar Multiplication, and Transposition

Linear Algebra

Matrix Addition

Matrix :drill: Scheduled: 2024-04-15 [A matrix] is a rectangular array of numbers.

Sizes

In general, a matrix of size mxn is of the form:

$\begin{array}{r}A=\left[\begin{array}{cccc}{a}_{1,1}& a_{1,2}& ...& a_{1,n}\\ a_{2,1}& a_{2,2}& ...& a_{2,n}\\ ...& & & \\ a_{m,1}& a_{m,2}& ...& a_{m,n}\end{array}\right]\end{array}$

This is denoted as $A=\left[\begin{array}{c}{a}_{ij}\end{array}\right]$.

A :drill: Scheduled: 2024-04-15 Ex:

$A=\left[\begin{array}{ccc}1& 2& -1\\ 0& 5& 6\end{array}\right]$

A is of size [2x3 (rows x columns)||dimension].

B :drill: Scheduled: 2024-04-08 $B=\left[\begin{array}{cc}1& -1\\ 0& 2\end{array}\right]$

B is [2x2||dimension].

C :drill: Scheduled: 2024-04-15

$\begin{array}{r}C=\left[\begin{array}{c}-1\\ -19\\ 17\end{array}\right]\end{array}$

C is [3x1||dimension].

Square matrices :drill: Scheduled: 2024-04-08 Matrices of nxn are called [square] matrices.

Equality :drill: Scheduled: 2024-04-08 Two matrices $A=[a_{ij}]$ and $B=[b_{ij}]$ are called [equal] if $a_{ij}=b_{ij}$ for all i and j.

Summation

If $A=[a_{ij}]$ and $B=[b_{ij}]$ are of the same size, their sum is

$\begin{array}{r}A+B=\left[\begin{array}{c}{a}_{ij}+b_{ij}\end{array}\right]\end{array}$

They have to be the same size.

Notes

1. Addition is not defined for matrices of different sizes.
2. If A,B and C are of the same size, then $A+B=B+A$ and $A+(B+C)=(A+B)+C$.

A zero matrix will be denoted as 0.

$\begin{array}{r}0+A=A\\ A-A=0\end{array}$

Scalar Multiplication

Definition

If $A=[a_{ij}]$ and k is a number, then $kA=[k{a}_{ij}]$ Which is identical to multiplying each entry of A by k.

Example

If kA=0, show that either k=0 or A=0.

Proof: Write $A=[a_{ij}]$ so that kA=0, means $k{a}_{ij}=0$.

For all i and j:

• If $k=0$, there is nothing to prove.
• if $k\ne 0$, and $k{a}_{ij}$ = 0, then $a_{ij}=0$ for all i and j (i.e. A=0)

Some properties A,B,C re matrices of size mxn, k & p are real numbers.

1. a+b = b+a
2. a+(b+c) = (a+b)+c
3. 0+A = A for each A.
4. a+(-A) = 0
5. k(A+B) = kA+kB
6. (k+p)A = kA + pA
7. (kp)A = k(pA)
8. 1A = A

Transpose of a matrix

Definition :drill: Scheduled: 2024-04-08 If A is an mn matrix, then transpose of A written as $A^T$, is the nm matrix where the [rows are the columns of A in the same order].

Example

Write down the transpose of each of the following matrices:

B = \begin{bmatrix} 5&2&6 \end{bmatrix} C = \begin{bmatrix} 1&2\\ 3&4\\ 5&0\\ \end{bmatrix} D = \begin{bmatrix} 3 & 1 & -1\\ 1 & 3 & 2\\ -1 & 2 & 1 \end{bmatrix} \end{aligned} $$ Solution $$ \begin{aligned} A = [1 3 2]\\ B = [5\\ 2\\ 6]\\ C = [1 3 5\\ 2 4 0]\\ D = D^T \end{aligned} $$ ### Theorem 2.1.2 :drill: Scheduled: 2024-04-08 A & B are matrices of the same size and k is a real number, then: 1. If $A$ is an m*n matrix, then $A^T$ is [n*m||dimension]. 2. $(A^T)^T = A$ 3. $(kA)^T = kA^T$ 4. $(A+B)^T = A^T+B^T$ #### Proof 1 is proved "by definition" a transpose 2: let $A=[a_{ij}]$ then $[a_{ij}]^T = A^T = [a_{ji}]$ hence, $(A^T)^T = [a_{ji}]^T = [a_{ij}] = A$ 3: $$ A = [a_{ij}] kA = [ka_{ij}] (kA)^T = [ka_{ij}]^T = k \cdot [a_{ij}]^T = kA^T $$ 4: $$ A = [a_{ij}] B = [b_{ij}] $$ A+B = [c_{ij}], where c_{ij} = (a_{ij} + b_{ij}) Then, $$ (A+B)^T = (c_{ij})^T = [c_{ji}] = [a_{ji} + b_{ji}] = [a_{ji}] + [b_{ji}] = A^T+B^T $$ ### Def :drill: Scheduled: 2024-04-08 If $A=[a_{ij}]$ is an m*n matrix, the elements (on the diagonal) $a_{1,1},a_{2,2},..,etc$ are called the [main diagonal]. ### Def :drill: Scheduled: 2024-04-14 A matrix A is called [symmetric] if $A=A^T$. Ex: $$ \begin{bmatrix} 3&1&-1\\ 1&3&2\\ -1&2&1\\ \end{bmatrix} $$