Zwei - Level - Dekomposition

Michael Goerz, 20. Mai 2007

Es soll die Matrix

U=(1/2){{1,1,1,1},{1,I,-1,-I},{1,-1,1,-1},{1,-I,-1,I}};
%//MatrixForm

in eine Reihe von Zwei - Level - Matrizen zerlegt werden.

1

Finde (Subscript[U, 1])

U1={{((U[[1]][[1]]\[Conjugate])/(Sqrt[U[[1]][[1]]  U[[1]][[1]]\[Conjugate]+U[[2]][[1]] U[[2]][[1]]\[Conjugate]])),((U[[2]][[1]]\[Conjugate])/(Sqrt[U[[1]][[1]]  U[[1]][[1]]\[Conjugate]+U[[2]][[1]] U[[2]][[1]]\[Conjugate]])),0,0},{((U[[2]][[1]])/(Sqrt[U[[1]][[1]]  U[[1]][[1]]\[Conjugate]+U[[2]][[1]] U[[2]][[1]]\[Conjugate]])),((-U[[1]][[1]])/(Sqrt[U[[1]][[1]]  U[[1]][[1]]\[Conjugate]+U[[2]][[1]] U[[2]][[1]]\[Conjugate]])),0,0},{0,0,1,0},{0,0,0,1}};
%//MatrixForm

M=U1.U//FullSimplify;
%//MatrixForm

Finde (Subscript[U, 2])

U2={{((M[[1]][[1]]\[Conjugate])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[3]][[1]] M[[3]][[1]]\[Conjugate]])),0,((M[[3]][[1]]\[Conjugate])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[3]][[1]] M[[3]][[1]]\[Conjugate]])),0},{0,1,0,0},{((M[[3]][[1]])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[3]][[1]] M[[3]][[1]]\[Conjugate]])),0,((-M[[1]][[1]])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[3]][[1]] M[[3]][[1]]\[Conjugate]])),0},{0,0,0,1}}//FullSimplify;
%//MatrixForm

U2.M//FullSimplify;
M=%;
%//MatrixForm

Finde (Subscript[U, 3])

U3={{((M[[1]][[1]]\[Conjugate])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[4]][[1]] M[[4]][[1]]\[Conjugate]])),0,0,((M[[4]][[1]]\[Conjugate])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[4]][[1]] M[[4]][[1]]\[Conjugate]]))},{0,1,0,0},{0,0,1,0},{((M[[4]][[1]])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[4]][[1]] M[[4]][[1]]\[Conjugate]])),0,0,((-M[[1]][[1]])/(Sqrt[M[[1]][[1]]  M[[1]][[1]]\[Conjugate]+M[[4]][[1]] M[[4]][[1]]\[Conjugate]]))}}//FullSimplify;
%//MatrixForm

U3.M//FullSimplify;
M=%;
%//MatrixForm

Dies muss weiter zerlegt werden.

Finde (Subscript[U, 4])

U4={{1,0,0,0},{0,((M[[2]][[2]]\[Conjugate])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[3]][[2]] M[[3]][[2]]\[Conjugate]])),((M[[3]][[2]]\[Conjugate])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[3]][[2]] M[[3]][[2]]\[Conjugate]])),0},{0,((M[[3]][[2]])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[3]][[2]] M[[3]][[2]]\[Conjugate]])),((-M[[2]][[2]])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[3]][[2]] M[[3]][[2]]\[Conjugate]])),0},{0,0,0,1}};
%//MatrixForm

U4.M//FullSimplify;
M=%;
%//MatrixForm

Finde (Subscript[U, 5])

U5={{1,0,0,0},{0,((M[[2]][[2]]\[Conjugate])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[4]][[2]] M[[4]][[2]]\[Conjugate]])),0,((M[[4]][[2]]\[Conjugate])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[4]][[2]] M[[4]][[2]]\[Conjugate]]))},{0,0,1,0},{0,((M[[4]][[2]])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[4]][[2]] M[[4]][[2]]\[Conjugate]])),0,((-M[[2]][[2]])/(Sqrt[M[[2]][[2]]  M[[2]][[2]]\[Conjugate]+M[[4]][[2]] M[[4]][[2]]\[Conjugate]]))}}//FullSimplify;
%//MatrixForm

U5.M//FullSimplify;
M=%;
%//MatrixForm

Finde (Subscript[U, 6])

U6=Inverse[M]//FullSimplify;
%//MatrixForm

Probe

Sind die gefundenen Matrizen eine Dekomposition von U?

U1\[ConjugateTranspose] . U2\[ConjugateTranspose] . U3\[ConjugateTranspose].U4\[ConjugateTranspose].U5\[ConjugateTranspose].U6\[ConjugateTranspose]//FullSimplify;
%//MatrixForm

U//MatrixForm

Ergibt das Produkt der Matrizen die Einheitsmatrix?

U6.U5.U4.U3.U2.U1.U//FullSimplify;
%//MatrixForm