Definition: Two complex matrices and are said to be similar if a non-singular (invertible) matrix such that . If can be chosen to be a unitary then and are unitarily similar.
- Similar matrices are representations of the same linear operator on in two different bases.
- Unitarily similar matrices represent the same linear operator but in two different orthogonal bases.
- 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 is said to be normal if .
The Spectral Theorem: A normal matrix is unitarily similar to a diagonal matrix.
Things to be noted
- If a matrix is normal then it will remain normal under unitary similarity.
- An upper triangular matrix is normal iff it is diagonal.
Schur Basis: The Schur basis of a normal matrix is a basis consisting of eigenvectors of . Normal matrices are, therefore, matrices whose eigenvectors form an orthonormal basis for .
Reference: Ranjendra Bhatia and Rhadha Mohan, “Triangularization of Matrix”, Resonance, June 2000.