Diagonal matrix

Definition

Let $R$ be a ring and let $n \ge 0$ be an integer. A square matrix $D = (d_{ij})$ of order $n$ over $R$ is diagonal when $d_{ij} = 0$ for all $i \ne j$. For $d_1, \dots, d_n \in R$, the matrix $\operatorname{diag}(d_1, \dots, d_n)$ is the diagonal matrix with $d_{ii} = d_i$ for each $i$.

Remarks

The product of two diagonal matrices is diagonal: $\operatorname{diag}(a_1, \dots, a_n) \operatorname{diag}(b_1, \dots, b_n) = \operatorname{diag}(a_1 b_1, \dots, a_n b_n)$. Over a commutative ring, diagonal matrices commute with each other. The determinant of a diagonal matrix is the product of its diagonal entries. A matrix that is similar to a diagonal matrix is diagonalisable.

Explanations

L1 human

A diagonal matrix has numbers only on its main diagonal, from the top left to the bottom right. As a map, it stretches each coordinate axis by its own factor. Diagonal matrices are easy to add, multiply and raise to powers.