Identity matrix

An identity matrix is a square matrix whose diagonal entries are all equal to one and whose off-diagonal entries are all equal to zero.

Identity matrices play a key role in linear algebra. In particular, their role in matrix multiplication is similar to the role played by the number 1 in the multiplication of real numbers:

a real number remains unchanged when it is multiplied by 1;
a matrix remains unchanged when it is multiplied by the identity matrix.

Table of contents

Definition
Examples
Products involving the identity matrix
The identity matrix is idempotent
Symmetry

Definition

The following is a formal definition.

Definition Let $I$ be a $K \times K$ matrix. $I$ is an identity matrix if and only if $I_{ij} = 1$ when $i = j$ and $I_{ij} = 0$ when $i \neq = j$ .

Thus, entries whose row index $i$ and column index $j$ coincide (i.e., entries located on the main diagonal) are equal to $1$ . All the other entries are equal to $0$ .

When $K = 1$ , there is only one entry, and

I = I_{11} = 1

Examples

Some examples of identity matrices follow.

Example The $2 \times 2$ identity matrix is

I = 1001

Example The $3 \times 3$ identity matrix is

I = 100010001

Example The $4 \times 4$ identity matrix is

I = 1000010000100001

Products involving the identity matrix

A key property is that a matrix remains unchanged when it is multiplied by the identity matrix.

Proposition Let $A$ be a $K \times L$ matrix and $I$ the $K \times K$ identity matrix. Then,

I A = A

Proof

By the definition of matrix product, the $(k, l)$ -th entry of the product $I A$ is

^{A}^{B} (I A)_{k l} = j = 1 \sum K I_{k j} A_{j l} = I_{k k} A_{k l} = 1 \cdot A_{k l} = A_{k l}

where: in step $A$ we have used the fact that $I_{k j} = 0$ when $k \neq = j$ ; in step $B$ we have used the fact that $I_{k k} = 1$ ( $I_{k k}$ is on the main diagonal of $I$ ). Since

(I A)_{k l} = A_{k l}

for every $k$ and $l$ , $I A = A$ .

Proposition Let $A$ be a $K \times L$ matrix and $I$ the $L \times L$ identity matrix. Then,

A I = A

Proof

The proof is similar to the previous one:

(A I)_{k l} = j = 1 \sum L A_{k j} I_{j l} = A_{k l} I_{l l} = A_{k l} \cdot 1 = A_{k l}

The identity matrix is idempotent

A consequence of the previous two propositions is that

I^{2} = I I = I

and

I^{n} = I I^{n - 1} = I^{n - 1} = I I^{n - 2} = I^{n - 2} = \dots = I

In other words, any power of an identity matrix is equal to the identity matrix itself.

A matrix possessing this property (it is equal to its powers) is called idempotent.

Symmetry

Another important property of the identity matrix is that it is symmetric, that is, equal to its transpose:

I^{⊤} = I

Proof

A matrix $I$ is symmetric if and only if

(I^{⊤})_{ij} = I_{j i} = I_{ij}

for any $j$ and $i$ . But the above equality always holds when $i = j$ , and it holds for identity matrices when $i \neq = j$ because

I_{j i} = 0 = I_{ij} when i \neq = j

How to cite

Please cite as:

Taboga, Marco (2021). "Identity matrix", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/identity-matrix.