Simultaneous Matrix Diagonalization
Simultaneous diagonalization of a group of matrices plays an important role in identifying irreducible representations of a group from its regular representation. But before delving into simultaneous diagnolization of regular matrix represenation of all elements of a group, let us briefly discuss how a single matrix can be diagnonalized.
Diagnonalization of a Single Matrix
Let matrix \(A\) be a \(n \times n\) matrix, which has following eigenvalues:
- \(m\) distinct real eignevalues, \(\lambda_1, \lambda_2, \lambda_3, .... , \lambda_m\)
- One complex conjugate pair, \(\lambda + i\omega, \lambda - i\omega\)
- \(\mu\) with algebraic and geometric mutiplicity 2, \(s.t. (A- \mu I)\) has a 2-D NULL space.
- \(\gamma\) with algebraic multiplicity 2 and geometric mutiplicity 1, \(s.t. (A- \gamma I)\) has a 1-D NULL space.