Scalar matrix

Definition

Let $R$ be a ring, let $n \ge 0$ be an integer, and let $c \in R$. The scalar matrix of order $n$ with value $c$ is the diagonal matrix $\operatorname{diag}(c, \dots, c) = c I_n$, where $I_n$ is the identity matrix. A square matrix is a scalar matrix when it equals $c I_n$ for some $c \in R$.

Remarks

Over a commutative ring, a scalar matrix commutes with every square matrix of the same order. Over a field, the scalar matrices are exactly the $n \times n$ matrices that commute with every $n \times n$ matrix. As a linear map, $c I_n$ multiplies each vector by $c$.

Explanations

L1 human

A scalar matrix has the same number at every place on the diagonal and zeros elsewhere. Multiplying by it is the same as multiplying by that number. For example, $3 I_2 = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix}$ triples each vector.