Eigendecomposition

Definition: Diagonalisable Matrix

A square matrix $A\in F^{n\times n}$ is diagonalisable if it is similar to a diagonal matrix $\Lambda \in F^{n\times n}$.

Theorem: Eigendecomposition

A square matrix $A\in F^{n\times n}$ is diagonalisable if and only if it has $n$ linearly independent eigenvectors ${\vec{e}}_1,\cdots ,{\vec{e}}_n$.

In that case, $A$ can be written as a matrix product

$$A=Q\Lambda Q^{-1},$$

where the $k$-th column of $Q$ is the ${\vec{e}}_k$ and $\Lambda =\mathrm{diag}({\lambda }_1,\cdots ,{\lambda }_n)$ is the diagonal matrix whose $k$-th diagonal entry is the eigenvalue ${\lambda }_k$ to which ${\vec{e}}_k$ belongs.

PROOF

TODO

Definition: Eigendecomposition

The eigendecomposition of $A$ is the product $Q\Lambda Q^{-1}$.