# Complex matrix problem

## Main Question or Discussion Point

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.

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

HallsofIvy
Homework Helper
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").

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