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Complex matrix problem

by broegger
Tags: complex, matrix
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broegger
#1
May17-04, 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.
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matt grime
#2
May17-04, 05:00 AM
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You can recover A form N, and if U diagonalizes N, does it diagonalize N*?
HallsofIvy
#3
May17-04, 05:01 AM
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Thanks
PF Gold
P: 39,564
Clearly, U*NU= U*AU+ i U*BU. Since U*NU is a diagonal matrix, all non-diagonal elements are 0. That is, All non-diagonal elements of U*AU and iU*BU must cancel. What does that tell you about them individually (and don't forget the "i").

broegger
#4
May17-04, 06:04 AM
P: 258
Complex matrix problem

Clearly, U*NU= U*AU+ i U*BU. Since U*NU is a diagonal matrix, all non-diagonal elements are 0. That is, All non-diagonal elements of U*AU and iU*BU must cancel. What does that tell you about them individually (and don't forget the "i").
I honorstly don't know... I don't think the fact that two matrices P and Q sum up to a diagonalmatrix D implies that they are diagonalmatrices themselves - it just means that their non-diagonal elements cancel - as you say yourself... Or what?
matt grime
#5
May17-04, 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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