* 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.