Lecture 4 Unitary Matrices

1 Lecture4 UnitaryMatrices MATRIXANALYSIS HITSZTIME Autumn2011INSTRUCTOR You HuaFan ReadingassignmentSection2 1 1 2 1 7Section2 2 1 2 2 2Section2 3 1 2 3 2Section2 4 1 2 4 4 2 Inthislecturewewillstudyaveryimportantclassofmatricesthatarequiteusefulinprovinganumberofstructuretheoremsaboutallmatrices calledunitarymatrices Asatransformation unitarymatrixpreserveslengthandpreservesanglesbetweenvectors Themodulusofeacheigenvalueofunitarymatrixisone Theseareofcoursetruefortheidentitytransformation Thereforeitishelpfultoregardunitarymatrixas generalizedidentity Inthislecture withoutexceptiontheunderlyinginnerproductisstandardinnerproductandtheunderlyingvectornormisEuclideannorm 2 3 4 1UnitaryMatricesandUnitarilyEquivalent 4 Exercise arerealorthogonal Theproductsofthemarealsorealorthogonal Inversely everyorthogonalmatrixcanbeinterpretedaseitherarotation areflection orsomecombinationofeitherofthese 5 Exercise IfUisunitary computeand U 2 Solution 6 7 Briefproof It seasytoproof a b c d e f areequivalentbecause For g h wehave 8 9 Briefproof Observation4 1 1 ThesetofallunitarymatricesinMnformangroup Inparticular theproductofunitarymatricesisunitary Also theproductoftwoorthogonalmatricesisorthogonal Briefproof 10 11 4 2Schur sTheorem Theorem4 2 1 Schur sTheorem EverymatrixAisunitarilyequivalenttoatriangularmatrix inwhichthediagonalentriesareeigenvaluesofA proof 12 13 14 4 3Cayley HamiltonTheorem 4 3 1 4 3 2 Briefproof 15 4 3 1 Briefproof 16 4 3 1 Briefproof Exercise UsingCayley Hamiltontheoremtosolvetheinverseof 17 CONCLUSION Basicconcepts unitarymatrixisometryunitarilyequivalent Importantprinciples propertiesofunitarymatricestwounitarilyequivalentmatriceshavethesameFrobeniusnormSchur stheoremCayley Hamiltontheorem 18 HOMEWORK Section2 1 3