矩阵分析讲义 Unitary matrix and normal matrix.pdf

Leture4:Unitarymatrixand

normalmatrix

MatrixAnalysis

Inthislecturewewillstudyaveryimportantclassof

matricesthatarequiteusefulinprovinganumberof

structuretheoremsaboutallmatrices,calledunitary

matrices.

Asatransformation,unitarymatrixpreserveslengthand

preservesanglesbetweenvectors.Themodulusofeach

eigenvalueofunitarymatrixisone.Theseareofcourse

truefortheidentitytransformation.Thereforeitishelpful

toregardunitarymatrixas“generalizedidentity”.

Inthislecture,withoutexceptiontheunderlyinginner

productisstandardinnerproductandtheunderlyingvector

normisEuclideannorm|·2.

4.1UnitaryMatricesandUnitarilyEquivalent

1010cossin

Exercise.I,R,(T)



0101sincos

arerealorthogonal.Theproductsofthemarealsoreal

orthogonal.Inversely,everyorthogonalmatrixcanbe

interpretedaseitherarotation,areflection,orsome

combinationofeitherofthese.

Exercise.IfUisunitary,computeandU

()U2.

Solution.

(|)max{|:()}UU1

*

max:()UUU

1

2

Briefproof.It’seasytoproof(a),(b),(c),(d),(e),(f)are