Complex Eigenvectors, How do I check?

[Unassuming]

I have come to a problem where I have Eigenvalues = 2,2i,-2i and my Eigenvectors have i's in them. I usually check my work using my calculator to perform the operation of,

$$S^{-1}AS=J$$

where S is my Eigenvector matrix, A is my original.

I then see what my J matrix looks like. It should look like my eigenvalues coming down the diagonal in the same order of my vectors that were in S.

My calculator doesn't accept the number i, in a matrix though! Is there another way to check my work here?

Thank you

 

[HallsofIvy]

If your operator or matrix is a, an eigenvalue is $\lambda$, and a corresponding eigenvector is $\vec{v}$ then you check it by seeing if $A\vec{v}= \lambda\vec{v}$.

If your calculator does matrix calcullations, and accepts complex numbers (the TI-85, for example, allows you to enter complex numbers as "(a, b)" for a+ bi), I would be very surprised if it did not allow you to use complex numbers in matrices. If the matrices are not worse than 3 by 3 or 4 by 4 you can always check $A\vec{v}= \lambda\vec{v}$ by hand.

 

[LMKIYHAQ]

I hal

 

[Unassuming]

I have the TI-83 which I do not think does Complex Numbers in Matrices. I was thinking about upgrading my calculator to a more powerful unit though.

Also, when I check $A\vec{v}= \lambda\vec{v}$.

I am checking that I have found the right Eigenvector, correct?

Also, thank you.

 

[HallsofIvy]

Yes, that is, after all, the definition of "eigenvector".