Compute the Eigenvalues and Eigenvectors

[swtjuice]

Homework Statement

Compute the Eigenvalues and Eigenvectors of A
A= [0 0 1;0 2 0;3 0 0]

Homework Equations

|A-lamda*I|=0
where I know the lamdas and plug them into the above equation and expand the system of equations.

The Attempt at a Solution

I have solved for the eigenvalues and got +sqrt(3), -sqrt(3) and 2.
I have solved for the eigenvectors associated with +sqrt(3), -sqrt(3), and checked them in matlab.
For lamda (eigenvalue) of sqrt(3) the eigenvector is
[ 1;0;sqrt(3)]

For lamda (eigenvalue) of -sqrt(3) the eigenvector is
[ -1;0;sqrt(3)]

But for when the eigenvalue is equal to 2 I come up to problems. where the 2nd row of my matrix is all zeroes. This confuses me because I have checked the vector in MATLAB and know it should be [0;1;0]. Which is impossible based on the 2nd row being all zeroes.

 

[mighty2000]

Can someone explain why this is the case?



It is possible for the eigenvector to have a row of zeroes, as long as the other rows are not all zeroes and the vector is still linearly independent. In this case, the eigenvector for the eigenvalue of 2 is [0;1;0], as you have correctly found. This simply means that the eigenvector is a scalar multiple of the identity matrix, which is still a valid eigenvector.