Thread: Eigenvalues and Eigenvectors of a 3x3 matrix

1. Eigenvalues and Eigenvectors of a 3x3 matrix

I have to find 4 things for the Matrix A which is a 3x3 matrix with all values equal to 1
A=
1 1 1
1 1 1
1 1 1

a.) Without having to make extensive calculations explain why 0 is an eigenvalue of A
b.) Find 2 linearly independent Eigenvectors for the Eigenvalue 0
c.) The e-value 0 has both geometric and algebraic multiplicity 2. Find the remaining eigenvalue and associated eigenvector without extensive calculations.
d.) Determine the matrices S and Λ and show A S = S Λ

I've uploaded a picture of my work since I'm not too familiar with it all up.

a.) I know that there are 2 eigenvalues (0 and 3) because of my calculations above but I'm not sure how to explain that without extensive calculations. I think it has to do with the trace?

b.) I also know that two linearly independent eigenvectors are v1 = (-1,0,1) and v2 = (-1,1,0) but I'm not sure how to show this.

c.) I think the remaining Eigenvalue is just 3, but I knew that from before hand when I found the e-values the long way. And it's associated e-vector would just be (1,1,1) I'm not sure how to show this in a non extensive calculations way.

and as for d.) I don't really know

2. Re: Eigenvalues and Eigenvectors of a 3x3 matrix

Hey grandmarquis84.

With regards to a) the determinant is the product of the eigenvalues. Since your determinant has to be zero (all rows are the same) then one eigenvalue must be 0.

For d) Consider the eigenvector/eigenvalue relationship where Ax = Λx for eigen-vectors x and eigen-value lambda. (You can also consider x as a compound matrix of all eigenvectors).