Solving Complex Matrix Problems: Normality, Diagonality, and Unitary Matrices

[broegger]

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.

 

[matt grime]

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

 

[HallsofIvy]

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]

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]

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