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Complex matrix problem 
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#1
May1704, 02:13 AM

P: 258

Let A and B be Hermitian matrices with AB = BA and let N = A + iB.
1) Show that N is normal. 2) Show that A = 1/2(N+N*) (* = conjugate transpose) and find a formula for B. 3) Let U be a unitary matrix such that U*NU is a diagonal matrix. Show that U*AU and U*BU is diagonal matrices. I had no problems with 1) and 2) but I simply can't figure out 3)... Please help. 


#2
May1704, 05:00 AM

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You can recover A form N, and if U diagonalizes N, does it diagonalize N*?



#3
May1704, 05:01 AM

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Clearly, U*NU= U*AU+ i U*BU. Since U*NU is a diagonal matrix, all nondiagonal elements are 0. That is, All nondiagonal elements of U*AU and iU*BU must cancel. What does that tell you about them individually (and don't forget the "i").



#4
May1704, 06:04 AM

P: 258

Complex matrix problem



#5
May1704, 06:14 AM

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I don't think Hall's method works since it doesn't use at any point the properties of A, B and N, and would thus appear to be 'true' for all matrices, which isn't possible.
However, U*NU diagonal implies (U*NU)*=U*N*U is diagonal, and you may recover U*AU from these two diagonal matrices using part 2 


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