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 iswhere: in step we have used the fact that when ; in step we have used the fact that ( is on the main diagonal of ). Sincefor 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 iffor any and . But the above equality always holds when , and it holds for identity matrices when because