I read from a book and claim that for any hermitian matrix can be diagonalized by a unitary matrix whose columns represent a complete set of its normalized eigenvectors. It then given an equation:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\mathbf{\left|A\right|=\left|U\right|\left|\Lambda\right|\left|U^{\dagger}\right|}=\lambda_{1}\lambda_{2}\ldots\lambda_{M} \qquad \qquad (1)[/tex]

Let [tex]\mathbf{A}[/tex] be a [tex]M\times M [/tex] hermitian matrix , and [tex]\lambda_{i}, \lambda_{2}, \ldots, \lambda_{M}[/tex] and [tex]\mathbf{u_{1},\, u_{2},\,\ldots\, u_{M}} [/tex]represent its eigenvalues and an orthonormal set of eigenvectors.

[tex]\mathbf{\Lambda=}diag\left[\lambda_{1},\lambda_{2},\ldots,\lambda_{M}\right][/tex]

and

[tex]\mathbf{U=[u_{1},u_{2},\ldots,u_{M}]}[/tex]

[tex]\mathbf{U^{\dagger}}[/tex] represent the complex conjugate transpose of U.

I failed to proof it in MATLAB.

I have

A =

1.5000 2.5000 3.5000 4.5000 5.5000

2.5000 1.5000 2.5000 3.5000 4.5000

3.5000 2.5000 1.5000 2.5000 3.5000

4.5000 3.5000 2.5000 1.5000 2.5000

5.5000 4.5000 3.5000 2.5000 1.5000

>> [U lamda] = eig(A)

U =

0.6015 -0.4703 -0.3717 0.1777 0.4973

0.3717 0.2490 0.6015 -0.5125 0.4187

-0.0000 0.6586 0.0000 0.6414 0.3936

-0.3717 0.2490 -0.6015 -0.5125 0.4187

-0.6015 -0.4703 0.3717 0.1777 0.4973

lamda =

-5.2361 0 0 0 0

0 -1.6080 0 0 0

0 0 -0.7639 0 0

0 0 0 -0.5558 0

0 0 0 0 15.6638

The middle of Equation (1) equal to

abs(U)*abs(lamda)*abs(U')

ans =

6.2463 4.8416 3.6267 4.8416 6.2463

4.8416 3.9914 3.0274 3.9914 4.8416

3.6267 3.0274 3.3521 3.0274 3.6267

4.8416 3.9914 3.0274 3.9914 4.8416

6.2463 4.8416 3.6267 4.8416 6.2463

it does not equal to the right handside of Equation 1:

prod(diag(lamda))

ans =

56.0000

I think I must made a stupid error somewhere but can't see it myself at the moment, someone can help me

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# Eigenvalue question, hermitian matrix

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