 StatLect

# 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. ## Definition

The following is a formal definition.

Definition Let be a matrix. is an identity matrix if and only if when and when .

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

When , there is only one entry, and ## Examples

Some examples of identity matrices follow.

Example The identity matrix is Example The identity matrix is Example The identity matrix is ## Products involving the identity matrix

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

Proposition Let be a matrix and the identity matrix. Then, Proof

By the definition of matrix product, the -th entry of the product is where: in step we have used the fact that when ; in step we have used the fact that ( is on the main diagonal of ). Since for every and , .

Proposition Let be a matrix and the identity matrix. Then, Proof

The proof is similar to the previous one: ## The identity matrix is idempotent

A consequence of the previous two propositions is that and 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: Proof

A matrix is symmetric if and only if for any and . But the above equality always holds when , and it holds for identity matrices when because 