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Eigenvalues and Eigenvectors of a Hermitian Matrix

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Please see the attached file for the full problem statement, and please show all steps in your solution.

Consider the Hermitian matrix

omega = 1/2
[2 0 0
0 3 -1
0 -1 3]

(1) Show that omega1 = omega2 = 1; omega3 = 2.

(2) Show that |omega = 2> is any vector of the form

1/((2a^2)^(1/2)) [0 a -a]

(3) Show that the omega = 1 eigenspace contains all vectors of the form

1/((b^2 + 2x^2)^(1/2)) [b c c]

either by feeding omega = 1 into the equations or by requiring that the omega = 1 eigenspace be orthogonal to |omega = 2>.

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The solution describes how to find the eigenvalues and eigenvector of a 3 x 3 Hermitian matrix.

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The characteristic polynomial of is:

The eigenvalues satisfy:

Thus we get:

So the eigenvalues are (with indices now assigned sequentially):

To find the ...

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