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.