geometric multiplicity= algebraic multiplicity for a symmetric matrix

Could any one tell me how to prove:

λ be an eigen value of a symmetric matrix A then how to show that the geometric multiplicity and algebraic multiplicity are equal?

Answer

Every symmetric matrix is diagonalizable (this can be proved by small perturbation argument), that is: it has a full set of orthogonal eigenvectors and is conjugate to a diagonal matrix. So, you only need to prove the statement for diagonal matrix. Symmetric matrices have no Jordan block in their spectral decomposition, that cause discrepancy in the geometric and algebraic multiplicities of eigenvalues.

Attribution
Source : Link , Question Author : Marso , Answer Author : DVD

Leave a Comment