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.
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
Some examples of identity matrices follow.
Example The identity matrix is
Example The identity matrix is
Example The identity matrix is
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,
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,
The proof is similar to the previous one:
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.
Another important property of the identity matrix is that it is symmetric, that is, equal to its transpose:
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
Most of the learning materials found on this website are now available in a traditional textbook format.