## Diagonalization

Diagonalization is the process of tranforming a square matrix $A$ to the diagonal form

Diagonal Form

- $T$ is the
*diagonal form* of $A$ if
- $T$ is Diagonal and
- there exists $X$ such that $T = X^{-1} A X$
- $A$ and $T$ are similar
- so $A$ and $T$ share the same eigenvalues

Non-defectiveness

- $A$ is
*non defective* $\iff$ there exists non singular $X$ s.t.
- $X^{-1} A X = \text{diag}(\lambda_1, ..., \lambda_n)$
- i.e. there exists a similarity tranformation $X$ such that the results is a diagonal matrix with eigenvalues on the diagonal

Eigendecomposition decomposes a symmetric matric $A$ as

- $\Lambda = S^{T} A S$
- where $\Lambda = \text{diag}(\lambda_1, ..., \lambda_n)$
- and $S$ has eigenvectors on the diagonal
- so the similarity transformation matrix $S$ here is orthogonal

## Sources