Identity matrix

An identity matrix for a specific operation is a matrix that, when combined with another compatible matrix using that operation, leaves the other matrix unchanged.

That’s a lot of words, so what I mean is that, if we have a matrix \mathbf{A} and an identity matrix \mathbf{I} of matching, compatible, dimensions, then:

\mathbf{A} \oplus \mathbf{I} = \mathbf{A}

\mathbf{I} \oplus \mathbf{A} = \mathbf{A}

(Where \oplus is whatever operation we’re talking about, like matrix addition or matrix multiplication).

Identity matrix for addition

For addition, we need to add nothing to a matrix to get the same matrix back. So the identity matrix for addition is a matrix where all the elements are zero, which we (imaginatively) call the zero matrix.

Identity matrix for multiplication

The identity matrix for multiplication is a matrix where all the elements on the leading diagonal are 1, and all the other elements are 0.

This is generally what people are talking about when they say ‘the identity matrix’, and we write it as \mathbf{I}.

See matrix multiplication identity for more details on the identity for matrix multiplication.