# ML Wiki

## 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

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