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!

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- Oct 29th 2008, 06:34 PMbambyEigenvalues
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! - Oct 29th 2008, 10:25 PMHallsofIvy
An eigenvalue of a matrix, A, is a value $\displaystyle \lambda$ such that $\displaystyle Av= \lambda v$ for some non-zero vector v. That is the same as $\displaystyle Av- \lambda v= (A- \lambda I)v= 0$. Since v= 0 is an obvious solution to that problem, we are saying that this must not have a unique solution and so $\displaystyle A- \lambda I$ 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

$\displaystyle \left|\begin{array}{ccc}1-\lambda & 0 & 0 \\ 1 & 1-\lambda & 1 \\ 1 & -1 & 1-\lambda \end{array}\right|= 0$

Expanding on the first row, that is

$\displaystyle (1- \lambda)\right|\begin{array}{cc}1-\lambda & 1 \\ -1 & 1-\lambda \end{array}\right|= 0$

$\displaystyle (1-\lambda)((1-\lambda)^2+ 1= (1-\lambda)^3+ (1-\lambda)= 0$

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? - Oct 30th 2008, 07:11 PMbambyEigenvalues
http://www.mathhelpforum.com/math-he...981ad2e5-1.gif

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!