Diagonal Matrices#
Definition: Diagonal Matrix
A diagonal matrix is a square matrix \(M \in F^{n \times n}\) which has non-zero entries only on its diagonal.
\[ \begin{bmatrix}\mu_1 & \cdots & 0_F \\ \vdots & \ddots & \vdots \\ 0_F & \cdots & \mu_n\end{bmatrix} \]
Notation
\[ \operatorname{diag}(\mu_1, \cdots, \mu_n) \]
Theorem: Inverting a Diagonal Matrix
A diagonal matrix \(\operatorname{diag}(\mu_1, \cdots, \mu_n)\) is invertible if and only if none of its diagonal entries are zeroes.
If \(\operatorname{diag}(\mu_1, \cdots, \mu_n)\) is invertible, then its inverse is \(\operatorname{diag}(\mu_1^{-1}, \cdots, \mu_n^{-1})\).
Proof
TODO