# Definition of diagonal matrix

Put content here.**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 $i \neq j$.
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$$D = \begin{pmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{pmatrix}$$
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**Example:**
$$D = \begin{pmatrix} 3 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 5 \end{pmatrix}$$
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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: $\det(D) = d_1 \cdot d_2 \cdots d_n$

# Parents

* Basic terminology and notation