# MATB24H3 Lecture Notes - Diagonalizable Matrix, Diagonal Matrix, Unitary Matrix

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Published on 21 Apr 2013

Department

Mathematics

Course

MATB24H3

Professor

University of Toronto at Scarborough

Department of Computer & Mathematical Sciences

MAT B24S Fall 2011

MAT B24

Lecture 21

We have proven that all Hermitian matrices are unitarily diagonalizable.

Questions arise now as to which other questions have this property: Are all

matrices unitarily diagonalizable? Are the Hermitian matrices the only ones

that are unitarily diagonalizable?

EXAMPLE: Determine if A=

1 0 i

010

0 0 i

is unitarily diagonalizable

solution.

det(A−λI) = det

1−λ0i

0 1 −λ0

0 0 i−λ

= (1 −λ)2(i−λ)

There are two eigenvalues: λ1= 1 of algebraic multiplicity 2 and λ2=i

of algebraic multiplicity 1.

Eλ1=nullspace

0 0 i

0 0 0

0 0 i−1

=sp

1

0

0

,

0

1

0

so, λ1has geometric multiplicity 2=alg. multiplicity.

Eλ2=nullspace

1−i0i

0 1 −i0

0 0 0

=nullspace

1 + i0−1

0 1 0

0 0 0

=sp

1

0

1 + i

1

## 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.