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.

$$\left[\begin{array}{ccc}{\mu }_1& \cdots & 0_F\\ ⋮& \ddots & ⋮\\ 0_F& \cdots & {\mu }_n\end{array}\right]$$

Notation

$$\mathrm{diag}({\mu }_1,\cdots ,{\mu }_n)$$

Theorem: Inverting a Diagonal Matrix

A diagonal matrix $\mathrm{diag}({\mu }_1,\cdots ,{\mu }_n)$ is invertible if and only if none of its diagonal entries are zeroes.

If $\mathrm{diag}({\mu }_1,\cdots ,{\mu }_n)$ is invertible, then its inverse is $\mathrm{diag}({\mu }_1^{-1},\cdots ,{\mu }_n^{-1})$.

PROOF

TODO