Definition of diagonal matrix

Definition: A diagonal matrix is a square matrix in which all off-diagonal entries are zero. That is, $D = (d_{ij})$ is diagonal if $d_{ij} = 0$ whenever $i \neq j$.

$$D = \begin{pmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{pmatrix}$$

Example:
$$D = \begin{pmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5 \end{pmatrix}$$

Diagonal matrices are particularly easy to work with:

• Matrix multiplication of diagonal matrices is commutative
• Powers: $D^k = \text{diag}(d_1^k, d_2^k, \ldots, d_n^k)$
• Inverse (when all $d_i \neq 0$): $D^{-1} = \text{diag}(1/d_1, 1/d_2, \ldots, 1/d_n)$
• Determinant: $\det(D) = d_1 \cdot d_2 \cdots d_n$