**Similarity transformation** A matrix

**A** and

**B** are called similar when

**B **=

**P**^{-1}AP Matrix

**P**^{-1}AP is the similarity transformation of A.

Orthogonal transformation: When A is orthogonal then

**P**^{-1}AP is a orthogonal transformation.

Unitary transformation: When A is unitary then

**P**^{-1}AP is a unitary transformation.

Implications for eigen values and eigen vectors:

The eigenvalues of A and B are the same.

Each eigenvector w of B can be obtained from an eigevector v of A: w=

**P**^{-1}v

Note that A, B and P are matrices and v and w are vectors.

__Source:__ http://en.wikipedia.org/wiki/Matrix_similarity