1. ## Eigenvalues

Hi, can you help me with this:
Define T is in F^3 by TX=AX, where
A= (1 0 0
1 1 1
1 -1 1);
i) If F=R (reals), determine all eigenvalues and eigenvectors of T;
i) If F=C (complexes), determine all eigenvalues and eigenvectors of T;
Thank you!

2. Originally Posted by bamby
Hi, can you help me with this:
Define T is in F^3 by TX=AX, where
A= (1 0 0
1 1 1
1 -1 1);
i) If F=R (reals), determine all eigenvalues and eigenvectors of T;
i) If F=C (complexes), determine all eigenvalues and eigenvectors of T;
Thank you!
An eigenvalue of a matrix, A, is a value $\lambda$ such that $Av= \lambda v$ for some non-zero vector v. That is the same as $Av- \lambda v= (A- \lambda I)v= 0$. Since v= 0 is an obvious solution to that problem, we are saying that this must not have a unique solution and so $A- \lambda I$ must not have an inverse. A condition that it not have an inverse is that its determinant is 0. So we have the "characteristic equation" which, for this matrix, is
$\left|\begin{array}{ccc}1-\lambda & 0 & 0 \\ 1 & 1-\lambda & 1 \\ 1 & -1 & 1-\lambda \end{array}\right|= 0$
Expanding on the first row, that is
$(1- \lambda)\right|\begin{array}{cc}1-\lambda & 1 \\ -1 & 1-\lambda \end{array}\right|= 0$
$(1-\lambda)((1-\lambda)^2+ 1= (1-\lambda)^3+ (1-\lambda)= 0$
i) What are the real roots of that? Can you find the eigenvector?

ii) What are the complex roots of that? can you find the eigenvectors?

3. ## Eigenvalues

i) What are the real roots of that? Can you find the eigenvector?
1 is the real root, (a,b,c)=(0,0,0) is eigenvector
ii) What are the complex roots of that? can you find the eigenvectors?
1+2i, 1-2i are the complex roots, x=(0, L, 2iL), x=(0, L, -2iL) are eigenvectors

Is that right? I am not sure about (a,b,c)=(0,0,0)

Thank you!