MATB24H3 Lecture Notes - Diagonalizable Matrix, Diagonal Matrix, Unitary Matrix
Document Summary
We have proven that all hermitian matrices are unitarily diagonalizable. 0 1 0 solution. det(a i) = det . = (1 )2(i ) i. There are two eigenvalues: 1 = 1 of algebraic multiplicity 2 and 2 = i of algebraic multiplicity 1. 0 so, 1 has geometric multiplicity 2=alg. multiplicity. 1 + i and 2 has geom mult. Thus a is diagonalizable and if c = . Notice however that e 1 and e 2 are not orthogonal. + = 1 6= 0 thus this non hermitian matrix is not unitarily diagonalizable. So: not all diagonalizable matrices are unitarily diagonalizable. The answer is in the theorem that follows. Definition: a square complex matrix a is normal if aa = a a. Theorem: a square complex matrix a is unitarily diagonalizable if and only if it is normal ie. aa = a a proof. (= ) we assume that a is unitarily diagonalizable.
