Vectors and matrices

This lecture provides an informal introduction to matrices and vectors.

Table of contents

Matrix
Dimension of a matrix
Entries of a matrix
Vectors
Scalars
Equal matrices
Zero matrices
Square matrices
Diagonal and off-diagonal elements
Identity matrix
Transpose of a matrix
Symmetric matrices
Solved exercises

Matrix

A matrix is a two-dimensional array that has a fixed number of rows and columns and contains a number at the intersection of each row and column.

A matrix is usually delimited by square brackets.

Example Here is an example of a matrix having two rows and two columns:

1431

Dimension of a matrix

If a matrix has $K$ rows and $L$ columns, we say that it has dimension $K \times L$ , or that it is a $K \times L$ matrix.

Example The matrix

A = 125901

has $2$ rows and $3$ columns. So, we say that $A$ is a $2 \times 3$ matrix.

Entries of a matrix

The numbers contained in a matrix are called entries of the matrix (or elements, or components).

If $A$ is a matrix, the entry at the intersection of row $k$ and column $l$ is usually denoted by $A_{k, l}$ (or $A_{k l}$ ). We say that it is the $(k, l)$ -th entry of $A$ .

Example Let $A$ be a $3 \times 2$ matrix defined as follows:

A = 178253

The element of $A$ at the intersection of the third row and the first column, that is, its $(3, 1)$ -th entry is

A_{3, 1} = 8

Vectors

If a matrix has only one row or only one column it is called a vector.

A matrix having only one row is called a row vector.

Example The $1 \times 3$ matrix

A = [121]

is a row vector because it has only one row.

A matrix having only one column is called a column vector.

Example The $2 \times 1$ matrix

A = 54

is a column vector because it has only one column.

Scalars

A matrix having only one row and one column is called a scalar.

Example The $1 \times 1$ matrix

A = [3]

is a scalar. In other words, a scalar is a single number.

Equal matrices

Equality between matrices is defined in the obvious way.

Two $K \times L$ matrices $A$ and $B$ having the same dimension are said to be equal if and only if all their corresponding elements are equal to each other:

A_{k l} = B_{k l} for k = 1, \dots, K and l = 1, \dots, L

Zero matrices

A matrix $A$ is a zero matrix if all its elements are equal to zero, and we write

A = 0

Example If $A$ is a $2 \times 3$ matrix and $A = 0$ , then

A = 000000

Square matrices

A $K \times L$ matrix is called a square matrix if the number of its rows is the same as the number of its columns, that is, $K = L$ .

Example The $2 \times 2$ matrix

A = 3341

is a square matrix.

Example The $3 \times 3$ matrix

A = 561 0 \frac{1}{2} 0 \frac{1}{3} 40

is a square matrix.

Diagonal and off-diagonal elements

Let $A$ be a square matrix.

The diagonal (or main diagonal of $A$ ) is the set of all entries $A_{k, l}$ such that $k = l$ .

The elements belonging to the diagonal are called diagonal elements, and all the other entries are called off-diagonal.

Example Let $A$ be the $3 \times 3$ matrix defined by

A = 533323331

All off-diagonal entries of $A$ are equal to $3$ , while the three diagonal elements are equal to $5$ , $2$ , and $1$ , respectively.

Identity matrix

A square matrix is called an identity matrix if all its diagonal elements are equal to $1$ and all its off-diagonal elements are equal to $0$ . It is usually indicated by the letter $I$ .

Example The $3 \times 3$ matrix

I = 100010001

is the $3 \times 3$ identity matrix.

Transpose of a matrix

If $A$ is a $K \times L$ matrix, its transpose, denoted by $A^{⊤}$ , is the $L \times K$ matrix such that the $(l, k)$ -th element of $A^{⊤}$ is equal to the $(k, l)$ -th element of $A$ for any $k$ and $l$ satisfying $1 \leq k \leq K$ and $1 \leq l \leq L$ .

In other words, the columns of $A^{⊤}$ are equal to the rows of $A$ (equivalently, the rows of $A^{⊤}$ are equal to the columns of $A$ ).

Example Let $A$ be the $2 \times 3$ matrix defined by

A = 75 5 \frac{1}{2} 01

Its transpose $A^{⊤}$ is the following $3 \times 2$ matrix:

A^{⊤} = 750 5 \frac{1}{2} 1

Example Let $A$ be the $2 \times 2$ matrix defined by

A = 4153

Its transpose $A^{⊤}$ is the following $2 \times 2$ matrix:

A^{⊤} = 4513

Symmetric matrices

A square matrix is said to be symmetric if it is equal to its transpose.

Example Let $A$ be the $2 \times 2$ matrix defined by

A = 4553

Its transpose $A^{⊤}$ is the following $2 \times 2$ matrix:

A^{⊤} = 4553

which is equal to $A$ . Therefore, $A$ is symmetric.

Solved exercises

Below you can find some exercises with explained solutions.

Exercise 1

Let $A$ be a $3 \times 3$ matrix defined by

A = 311230100

Find its transpose.

Solution

The transpose $A^{⊤}$ is a matrix such that its columns are equal to the rows of $A$ :

A^{⊤} = 321130100

Exercise 2

Let $A$ be a $3 \times 1$ column vector defined by

A = 312

Show that its transpose is a row vector.

Solution

The transpose $A^{⊤}$ is a matrix such that its rows are equal to the columns of $A$ . But $A$ has only one column, which implies that $A^{⊤}$ has only one row. Therefore, it is a row vector:

A^{⊤} = [312]

Exercise 3

Let $A$ be a $3 \times 3$ matrix defined by

A = 211131100

Is it symmetric?

Solution

$A$ is symmetric if it is equal to its transpose. The transpose of $A$ is

A^{⊤} = 211130110

which is not equal to $A$ . Therefore, $A$ is not symmetric.

How to cite

Please cite as:

Taboga, Marco (2021). "Vectors and matrices", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/vectors-and-matrices.