- #1

jeebs

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[tex]A = \left(\begin{array}{ccc}0&i&2\\0&1&0\\1&0&0\end{array}\right) [/tex]and [tex]B = \left(\begin{array}{ccc}2&i&0\\3&1&5\\0&-i&-2\end{array}\right)[/tex]

Calculate A

^{-1}B andBA

^{-1}. Are they equal?

Hint: find the inverse of A in the diagonal basis.

I am really not sure how to proceed with this. The "diagonal basis" hint made me think I should be looking to rewrite A like

[tex]A = \left(\begin{array}{ccc}A_1_1&0&0\\0&A_2_2&0\\0&0&A_3_3\end{array}\right) [/tex]

I was thinking I need eigenvalues to put on the leading diagonal so I attempted to use an eigenvalue equation [tex]A|u> = \lambda|u> [/tex], where I start with [tex] |u> = \left(\begin{array}{c}u_1\\u_2\\u_3\end{array}\right)[/tex].

If I do this I end up with the system of simultaneous equations:

[tex]iu_2 + 2u_3 = \lambda u_1[/tex]

[tex]u_2 = \lambda u_2[/tex]

[tex]u_1 = \lambda u_3[/tex]

From the 2nd one I can say that [tex]\lambda = 1[/tex] which gives me:

[tex]u_1 = u_3[/tex] and so [tex]|u> = \left(\begin{array}{c}u_1\\-u_1/i\\u_1\end{array}\right) = \left(\begin{array}{c}1\\-1/i\\1\end{array}\right) = \left(\begin{array}{c}1\\i\\1\end{array}\right) [/tex] which is normalized.

This problem was part of a past problem class I went to, and on the board was written "Diagonalize matrix: A

_{D}= U

^{+}AU" (where the + is meant to be a dagger really, denoting an adjoint). Looking back now this expression means nothing to me, but also we were told that apparently we should end up with [tex]U = \left(\begin{array}{ccc}-\sqrt{2}&sqrt{2}&1\\0&0&i\\1&1&1\end{array}\right) [/tex] and [tex]A_D = \left(\begin{array}{ccc}-\sqrt{2}&0&0\\0&\sqrt{2}&0\\0&0&1\end{array}\right) [/tex] and that the columns of U are eigenvectors. I do not know what the hell to make of this, but I do notice that the eigenvector that I worked out in the stuff I have just shown above is the same as the third column, so I must be at least somewhere near the right track. I'm sorry I cannot explain more but this is everything I have to go on from what I wrote down.

However, I do not know:

a) how I am supposed to come up with 2 other eigenvectors

b) why, once I have 3 eigenvectors, I should arrange them into this 3x3 object called U

I get the feeling this is actually a straightforward task to do, I'm just not clear on the way to do it. Can anyone shed any light on this? It would be greatly appreciated. Thanks.