Eigenval/vect question, double check my answers, dont make sense

[imsleepy]

eigenval/vect question, double check my answers, don't make sense....

Homework Statement

find eigenvalues and eigenvectors of matrix A =
102
012
223

Homework Equations

A -λI etc blah

The Attempt at a Solution

I'm getting λ = 1, -1, 5.

For λ = 1, v = (-1,1,0)t (t being a real integer constant), and double checking says it's right.

For λ = -1, I'm getting v = (1,1,1)t but if i double check by multiplying it with the matrix A I'm not getting something in the form of v (i'm getting like.. 3,3,7, which doesn't make sense to me).

For λ = 5, i got v = (1,1,2)t but again double checking doesn't give a vector in this form! (multiplying this v by A gives me 3,3,14) :angry:

what am i doing wrong??

 

[ehild]

(1,1,1) is wrong, it is (-1,-1,1).
You do something wrong with matrix multiplication. How do you do it? The third eigenvector is correct.

ehild

 

[imsleepy]

ok yeah my bad, the evect for lambda = -1 is (-1,-1,1) (i rushed).

but multiplying A with (-1,-1,1) gives (-3,-3,7), which still doesn't make sense >:/

 

[lanedance]

i get (1,1,-1) as you would expect

 

[ehild]

How do you do the multiplication?

Multiply (-1,-1,1) with the
first column of the original matrix: -1+0+2=1;
with the second column: 0-1+2=1;
with the third column: -2-2+3=-1.
The result is (1,1,-1) =-1(-1,-1,1) as it has to be.

ehild

 

[imsleepy]

ok i was multiplying column of matrix by each row of the vector, when i should be multiplying the rows of the matrix by the vector and adding terms.

my bad.

 

[ehild]

It is the same, you can multiply each element of a row of the matrix with the corresponding element of the vector, and ADD them - so is it OK now?

ehild

 

[imsleepy]

A times (-1,-1,1) i (incorrectly) did:

1x-1 + 0x-1 + 2x-1
0x-1 + 1x-1 + 2x-1
2x 1 + 2x 1 + 3x 1

giving

-1-2
-1-2
2+2+3

resulting in (-3,-3,7)

which is incorrect.

i know how to do it now, thanks ehild and lanedance.
shouldve remembered this from years ago...