Vectors and Matrix Operations

Numbers, Vectors, Matrices

Recall that the set of real numbers is $\mathbb{R}$ and that a vector, $v \in \mathbb{R}^n$, is just an $n$-tuple of numbers. Similarly, an $n \times m$ matrix is just a table of numbers, with $n$ rows and $m$ columns and we can write

$A_{mn} \in \mathbb{R}^{mn}$

Note that a vector is normally considered equivalent to a $n\times 1$ matrix i.e. we view these as column vectors.

Examples

Example

In R, a vector can be generated with:

> X <- 3:6
> X
[1] 3 4 5 6

A matrix can be generated in R as follows,

> matrix(X)
     [,1]
[1,]    3
[2,]    4
[3,]    5
[4,]    6

Note

We note that R distinguishes between vectors and matrices.

Elementary Operations

We can define multiplication of a real number $k$ and a vector $v=(v_1,\ldots,v_n)$ by $k\cdot v=(kv_1,\ldots,kv_n)$. The sum of two vectors in $\mathbb{R}^n$, $v=(v_1,\ldots,v_n)$ and $u=(u_1,\ldots,u_n)$ is defined as the vector $v+u=(v_1+u_1,\ldots,v_n+u_n)$. We can define multiplication of a number and a matrix and the sum of two matrices (of the same sizes) similarly.

Examples

Example

> A <- matrix(c(1,2,3,4), nr=2, nc=2)

> A
     [,1] [,2]
[1,]    1    3
[2,]    2    4

> B <- matrix(c(1,0,2,1), nr=2, nc=2)

> B
     [,1] [,2]
[1,]    1    2
[2,]    0    1

> A+B
     [,1] [,2]
[1,]    2    5
[2,]    2    5

The Tranpose of a Matrix

In R, matrices may be constructed using the matrix function and the transpose of $A$, $A^\prime$, may be obtained in R by using the t function:

> A <- matrix(1:6, nrow=3)

> t(A)
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]    4    5    6

Details

If $A$ is an $n \times m$ matrix with element $a_{ij}$ in row $i$ and column $j$, then $A^\prime$ or $A^T$ is the $m\times n$ matrix with element $a_{ij}$ in row $j$ and column $i$.

Examples

Example

Consider a vector in R

> x <- 1:4

> x
[1] 1 2 3 4

> t(x)
     [,1] [,2] [,3] [,4]
[1,]    1    2    3    4

> matrix(x)
     [,1]
[1,]    1
[2,]    2
[3,]    3
[4,]    4
> t(matrix(x))
     [,1] [,2] [,3] [,4]

Note

The first solution gives a $1 \times n$ matrix and the second solution gives a $n \times 1$ matrix.

Matrix Multiplication

Matrices $A$ and $B$ can be multiplied together if $A$ is an $n \times p$ matrix and $B$ is an $p\times m$ matrix. The general element $c_{ij}$ of $n\times m$, $C=AB$, is found by pairing the $i^{th}$ row of $C$ with the $j^{th}$ column of $B$, and computing the sum of products of the paired terms.

Details

Matrices $A$ and $B$ can be multiplied together if $A$ is a $n\times p$ matrix and $B$ is a $p\times m$ matrix. Given the general element $c_{ij}$ of $n \times m$ matrix, $C=AB$ is found by pairing the $i^{th}$ row of $C$ with the $j^{th}$ column of $B$, and computing the sum of products of the paired terms.

Examples

Example: Matrices in R

> A <- matrix(c(1,3,5,2,4,6),3,2)

> A
     [,1] [,2]
[1,]    1    2
[2,]    3    4
[3,]    5    6

> B <- matrix(c(1,1,2,3),2,2)

> B
     [,1] [,2]
[1,]    1    2
[2,]    1    3

> A%*%B
     [,1] [,2]
[1,]    3    8
[2,]    7   18
[3,]   11   28

More on Matrix Multiplication

Let $A$, $B$, and $C$ be $m\times n$, $n\times l$, and $l\times p$ matrices, respectively. Then we have

$(AB)C=A(BC)$

In general, matrix multiplication is not commutative, that is $AB\neq BA$.

We also have

$(AB)'=B'A'$

In particular, $(Av)'(Av)=v'A'Av$, when $v$ is a $n\times1$ column vector

More obvious are the rules

1. $A+(B+C)=(A+B)+C$

2. $k(A+B)=kA+kB$

3. $A(B+C)=AB+AC$

where $k\in\mathbb{R}$ and when the dimensions of the matrices fit.

Linear Equations

Details

General linear equations can be written in the form $Ax=b$.

Examples

Example

The set of equations

$2x+3y=4$

$3x+y=2$

can be written in matrix formulation as

$$\begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 4 \\ 2 \end{bmatrix}$$

i.e. $A\underline{x} = \underline{b}$ for an appropriate choice of $A, \underline{x}$ and $\underline{b}$.

The Unit Matrix

The $n\times n$ matrix

$$I = \left[ \begin{array}{cccc} 1 & 0 & \ldots & 0 \\ 0 & 1 & 0 & \vdots \\ \vdots & 0 & \dots & 0 \\ 0 & \ldots & 0 & 1 \end{array} \right]$$

is the identity matrix. This is because if a matrix $A$ is $n\times n$

then $A I = A$ and $I A = A$

The Inverse of a Matrix

If $A$ is an $n \times n$ matrix and $B$ is a matrix such that

$BA = AB = I$

then $B$ is said to be the inverse of $A$, written

$B = A ^{-1}$

Note that if $A$ is an $n \times n$ matrix for which an inverse exists, then the equation $Ax = b$ can be solved and the solution is $x = A^{-1} b$.

Examples

Example

If matrix $A$ is:

$$\begin{bmatrix} 2 & 3 \\ 3 & 1 \end{bmatrix}$$

then $A ^{-1}$ is:

$$\begin{bmatrix} \displaystyle\frac{-1}{7} & \displaystyle\frac{3}{7} \\ \displaystyle\frac{3}{7} & \displaystyle\frac{-2}{7} \end{bmatrix}$$