- #1

dingo_d

- 211

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## Homework Statement

So I have to find the eigenvalues and eigenvectors of

[tex]A=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)[/tex]

which is not that special and hard. I solve the characteristic equation and find the eigenvalues: [tex]\lambda_{1,2}=\pm 1[/tex]. So finding the eigenvectors is relatively simple, I just plug that back into characteristic eq and I get:

[tex]v_1=t\left(\begin{array}{c}1\\1\end{array}\right)[/tex]

and

[tex]v_2=t\left(\begin{array}{c}1\\-1\end{array}\right)[/tex].

So that's pretty simple? I can even normalize it and show that they are orthogonal. But!

There was one thing that kinda bugged me, and got my attention. Back on linear algebra class we said that t is some real parameter and we lived our lives happily ever after.

But my professor, now on QM asked this: how do we know that t is real? What if it's complex?

If it is complex then the normalization isn't that trivial.

So my question is: why is it real? Are we free to impose that on the parameter? Or is there some deeper math behind it all?