# A question about a new way to find eigenvectors that i noticed

1. Mar 2, 2008

### transgalactic

i got this question in which we are given the matrix T
and we need to find the eigenvalues and the independent spaces (i dont know what is independent space) of T^2 +2*T

the problem is that he started to solve the question as i would have solved it
but then he puts a big X on it and does something else
i cant understand it??(and he gets all the point for it)

it looks as if he skips the finding the roots of polinomial step
why???

http://img253.imageshack.us/my.php?image=img86091xg4.jpg

Last edited: Mar 2, 2008
2. Mar 2, 2008

### Dick

The solver appears to have realized that he didn't have to compute the eigenvalues of T^2+2*T since he already knew that the eigenvalues of T were +1 and -1, apparently from a previous problem. This let him immediately conclude the eigenvalues of T^2+2*T are 3 and -1. Once he knew the eigenvalues he substituted them in for lambda and seems to have read off the eigenvectors more or less by inspection. It's not a new way of computing eigenvalues.

3. Mar 2, 2008

### HallsofIvy

Staff Emeritus
If $\lambda$ is an eigenvalue of T, with eigenvector v, then Tv= $\lambda$v. From that, $(T^2+ 2T)v= T(T(v))+ 2T(v)= T(\lambda v)- 2\lambda= \lambda T(v)- 2\lambda= \lambda(\lambda v)- 2\lambda v= (\lambda^2- 2\lambda) v$.

In other words if $\lambda$ is an eigenvalue of T with eigenvector v, then $\lambda^2- 2\lambda$ is an eigenvalue of T2- 2T with eigenvector v.

It is easier to find the eigenvalues of T and then use that formula than to find the eigenvalues of T2- 2T directly.

Last edited: Mar 4, 2008
4. Mar 4, 2008

### transgalactic

ok i understood how you got the formula from the T^2 + 2T epression

what now??
how do i mix up the eigenvalues of T with this formula in order to get the new values
??

5. Mar 4, 2008

### HallsofIvy

Staff Emeritus
What do you mean by "mix up the eigenvalues of T" and what "new values" are you talking about?

If you mean "How do I go from the eigenvalues of T to the eigenvalues of T2- 2T?", that's exactly what I told you before.:

6. Mar 4, 2008

### transgalactic

correct me if i am wrong

x-eigenvalue of T
y-eigen value of the expression
y(x)=x^2-2*x

so if x=-1 then for that "old" eigen value we get y=3
and we do that proccess for every eigenvalue