# Triangularization of a Matrix

Definition: Two $n\times n$ complex matrices $A$ and $B$ are said to be similar if $\exists$ a non-singular  (invertible) matrix $S$ such that $B=S^{-1}AS$. If $S$ can be chosen to be a unitary then $A$ and $B$ are unitarily similar.

1. Similar matrices are representations of the same linear operator on $\mathbb{C}^{n}$ in two different bases.
2. Unitarily similar matrices represent the same linear operator but in two different orthogonal bases.
3. Similarity preserves the rank, determinant, trace and eigenvalues of a matrix.

Schur’s Threorem: Every matrix is unitarily similar to an upper triangular matrix.

Definition: A matrix $A$ is said to be normal if $AA^{\dagger}=A^{\dagger}A$.

The Spectral Theorem: A normal matrix is unitarily similar to a diagonal matrix.

Things to be noted

1. If a matrix is normal then it will remain normal under unitary similarity.
2. An upper triangular matrix is normal iff it is diagonal.

Schur Basis: The Schur basis of a normal matrix $A$ is a basis consisting of eigenvectors of $A$. Normal matrices are, therefore, matrices whose eigenvectors form an orthonormal basis for $\mathbb{C}^n$.

Reference: Ranjendra Bhatia and Rhadha Mohan, “Triangularization of Matrix”, Resonance, June 2000.