# eigenvalues and vectors

• Mar 6th 2007, 02:35 PM
mesterpa
eigenvalues and vectors
hi.
i need to find the eigenvalues and eigenvectors of the following matrix.

( 7 -6 )
( -6 -2 )

i worked out the eigenvalues to be -2 and 7. can anyone tell me if these are right. not sure how to get eigenvectors.
thanks for the help
• Mar 6th 2007, 02:53 PM
Jhevon
Quote:

Originally Posted by mesterpa
hi.
i need to find the eigenvalues and eigenvectors of the following matrix.

( 7 -6 )
( -6 -2 )

i worked out the eigenvalues to be -2 and 7. can anyone tell me if these are right. not sure how to get eigenvectors.
thanks for the help

sorry, but those are wrong:

recall that to find the eigenvalues, we find det((lambda)In - A)

where In is the identity matrix and A is the matrix you're working on. i will use L for lambda, since i cant bother typing lambda all the time.

For eigenvalues:

det(L*In - A) = 0

(L..........0)......-.......(7...........-6)
(0..........L)..............(-6..........-2)

= (L - 7.........6)
...(6.......... L+2)

det(L - 7.........6) = 0
....(6.......... L+2)
=> (L - 7)(L + 2) - 36 = 0
=> L^2 -5L - 50 = 0
=> (L + 5)(L - 10)=0

so L1 = -5, L2 = 10 .............these are the eigen values
• Mar 6th 2007, 02:55 PM
ThePerfectHacker
Quote:

Originally Posted by mesterpa
hi.
i need to find the eigenvalues and eigenvectors of the following matrix.

( 7 -6 )
( -6 -2 )

i worked out the eigenvalues to be -2 and 7. can anyone tell me if these are right. not sure how to get eigenvectors.
thanks for the help

I think you are wrong.
• Mar 6th 2007, 02:57 PM
Jhevon
Quote:

Originally Posted by mesterpa
hi.
i need to find the eigenvalues and eigenvectors of the following matrix.

( 7 -6 )
( -6 -2 )

i worked out the eigenvalues to be -2 and 7. can anyone tell me if these are right. not sure how to get eigenvectors.
thanks for the help

to get the eigenvectors, you use (L*In - A)x = 0 where x is a vector of with components x1, x2 ....

so for each eigenvector, you plug in its value into the matrix that we formed, solve for the unknowns and place them in a solutions vector, these vectors will be your eigen vectors