Definition of unit matrix

Definition: A unit matrix is another name for the identity matrix $I_n$ — the $n \times n$ square matrix with ones on the main diagonal and zeros elsewhere:

$$I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}$$

The term "unit matrix" emphasizes that it acts as the multiplicative identity in the ring of $n \times n$ matrices: $AI = IA = A$ for any $n \times n$ matrix $A$.

In some contexts, "unit matrix" may also refer to a matrix unit $E_{ij}$ — a matrix with a 1 in position $(i,j)$ and zeros elsewhere. These form a basis for the vector space of all $m \times n$ matrices.