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!
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 $\displaystyle \lambda$ 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?
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!