Matrix Addition, Scalar Multiplication, and Transposition

Linear Algebra

Matrix Addition

Matrix

[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

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

$$B=\left[\begin{array}{cc}1& -1\\ & \\ 0& 2\\ & \end{array}\right]$$

B is [2x2||dimension].

C

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

C is [3x1||dimension].

Square matrices

Matrices of nxn are called [square] matrices.

Equality

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

If A is an m*n matrix, then transpose of A written as $A^T$, is the n*m matrix where the [rows are the columns of A in the same order].

Example

Write down the transpose of each of the following matrices:

$$\begin{array}{r}A=\left[\begin{array}{c}1\\ \\ 3\\ \\ 2\end{array}\right]B=\left[\begin{array}{ccc}5& 2& 6\end{array}\right]C=\left[\begin{array}{cc}1& 2\\ & \\ 3& 4\\ & \\ 5& 0\\ & \end{array}\right]D=\left[\begin{array}{ccc}3& 1& -1\\ & & \\ 1& 3& 2\\ & & \\ -1& 2& 1\end{array}\right]\end{array}$$

Solution

$$\begin{array}{r}A=[132]\\ \\ B=[5\\ \\ 2\\ \\ 6]\\ \\ C=[135\\ \\ 240]\\ \\ D=D^T\end{array}$$

Theorem 2.1.2

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. $(kA{)}^T=k{A}^T$
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:

$$A=[a_{ij}]kA=[k{a}_{ij}](kA{)}^T=[k{a}_{ij}{]}^T=k\cdot [a_{ij}{]}^T=k{A}^T$$

4:

$$A=[a_{ij}]B=[b_{ij}]$$

A+B = [cij], where cij = (aij + bij)

Then,

$$(A+B{)}^T=(c_{ij}{)}^T=[c_{ji}]=[a_{ji}+b_{ji}]=[a_{ji}]+[b_{ji}]=A^T+B^T$$

Def

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

A matrix A is called [symmetric] if $A=A^T$.

Ex:

$$\left[\begin{array}{ccc}3& 1& -1\\ & & \\ 1& 3& 2\\ & & \\ -1& 2& 1\\ & & \end{array}\right]$$