Finding eigenvectors for a matrix

Need some help with this, the book is written in a very confusing way. I'm following the examples exactly and not getting the correct result.

The matrix A = [2,-1 ; -1,2]

I found the two eigenvalues, to be **λ **= 1 and **λ **= 3

Now when I do the A - **λ**I calculation and reduce the matrix I get

for **λ = 1**

x1 + x2 = 0 --> x1 = -x2

however, the solution in the book says the answer is

[1,1]

for **λ = 3**

-x1 + x2 = 0 --> x1 = x2

but again, the solution says

[-1, 1]

Can someone please clarify or point out what I am doing wrong?

Thanks.

Re: Finding eigenvectors for a matrix

So you solved the characteristic equation to get and . I got the same eigen values.

Now to find the eigen vectors

in which you get to solve a homogenous equation . Which yields infinetly many solutions of the form So one eigen vector for the eigen value would be p = 1,

For i got eigen vectors using the same processing above, infinitely many solutions of the form . So p = 1, is a eigen vector for eigen value

Re: Finding eigenvectors for a matrix

Quote:

Originally Posted by

**jakncoke** So you solved the characteristic equation to get

and

. I got the same eigen values.

Now to find the eigen vectors

in which you get to solve a homogenous equation

. Which yields infinetly many solutions of the form

So one eigen vector for the eigen value

would be p = 1,

For

i got eigen vectors using the same processing above, infinitely many solutions of the form

. So p = 1,

is a eigen vector for eigen value

I appologize, the matrix A is actually [2,1 ; 1,2]

could you check it once again with the right matrix

Re: Finding eigenvectors for a matrix

Re: Finding eigenvectors for a matrix

Quote:

Originally Posted by

**jakncoke** with this matrix the eigen vectors are merely exchanged.

for

i got eigen vectors of the form

so for p = 1

is the one of the eigen vector for

for

i got eigen vectors of the form

so for p = 1

is the one of the eigen vector for

That's exactly what I thought the correct answer was. But in the back of the book with the answers, it states that for eigenvalue 3, the vector is [-1,1] and that for eigenvalue 1, the vector is [1,1].

Could they have just switched them by accident? Because I am getting the same results as you are.

Re: Finding eigenvectors for a matrix

Yea, even books make mistakes