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Linear Algebra

  • Thread starter pivoxa15
  • Start date
  • #1
2,255
1
Find the diagonal form of the Hermitian matrix

[tex]A=\left(
\begin{array}{cc}
2 & 3i\\
-3i & 2
\end{array}
\right)
[/tex]

The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix.

I put the columns of P as the eigenvectors (with unit length) of A,

[tex]P=\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
i & -i\\
1 & 1
\end{array}
\right)
[/tex]

I have checked that P is unitary with [tex]P^{-1}=P^{*}[/tex] and the diagonal entries of D should be 5 and -1. But I got

[tex]D=\left(
\begin{array}{cc}
2 & -3\\
-3 & 2
\end{array}
\right)
[/tex]

which clearly isn't correct.
 
Last edited:

Answers and Replies

  • #2
matt grime
Science Advisor
Homework Helper
9,395
3
Yes, you're right it isn't correct. I'm, nots sure what you want anyone here to do. You have the method correct, so just make sure you'renot making any dumb mistakes in multiplying out matrices.
 
  • #3
76
0
hey
rigth method but wrong eigenvalues

P = 1/sqrt(2) [i -1;i 1]

this will help
 
  • #4
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
The spectral theorem could be used with PAP*=D where D is diagonal matrix and P is a unitary matrix.
I can never remember for sure, but isn't it supposed to be A = PDP*? (and thus P*AP = D?)
 
  • #5
2,255
1
greisen said:
hey
rigth method but wrong eigenvalues

P = 1/sqrt(2) [i -1;i 1]

this will help
That could be my mistake.
 

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