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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- October 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! - October 29th 2008, 10:25 PMHallsofIvy
An eigenvalue of a matrix, A, is a value such that for some non-zero vector v. That is the same as . Since v= 0 is an obvious solution to that problem, we are saying that this must not have a unique solution and so 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

Expanding on the first row, that is

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? - October 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!