identity matrix

The identity matrix is the matrix version of the number 1. Just as multiplying any number by 1 leaves it untouched, multiplying any matrix by the identity gives back the very same matrix. It is the “do nothing” of matrix multiplication.

It is a square matrix with 1s running down the main diagonal (top-left to bottom-right) and 0s everywhere else. We write it I, or I_n to show its size. The 2-by-2 identity is [1, 0; 0, 1]; the 3-by-3 has three 1s on the diagonal.

The defining property is that AI = A and IA = A whenever the sizes line up. Because of this, the identity matrix is the goal of inversion: a matrix B is the inverse of A exactly when AB = I. It is also the answer the identity element provides in the language of groups and rings.