xdrgnh said:Homework Statement
I'm given this matrice 2 1 0
1 2 0
0 0 3
and I need to find it's eigenvectors
Homework Equations
The Attempt at a Solution
So I get the eigenvalues to be 1,3,3 with 3 being the one with multiplicity of 2. For the eigenvector for 1 I get 1,-1,0 and for 3 I get 1,1,1 but here is the problem. For the other eigen vector for 3 the answer can be anything that satisfies x1=x2
x2=x1
x3=x3
so can something like 1,1,2 be the answer?
xdrgnh said:But wouldn't that mean there is no one definitive eigenbasis?
An eigenvector is a vector that when multiplied by a matrix, results in a scalar multiple of itself. In other words, the direction of the vector remains the same after the transformation.
A 2nd multiplicity eigenvalue is an eigenvalue that has a multiplicity of 2, meaning it has two eigenvectors associated with it. This is often seen in matrices with repeated eigenvalues.
Finding the eigenvectors of a 2nd multiplicity eigenvalue is important because it allows us to understand the behavior of a matrix and its transformations. It also helps in solving systems of differential equations and diagonalizing matrices.
To find the eigenvectors of a 2nd multiplicity eigenvalue, we first need to find the corresponding eigenvalue. Then, we can use this eigenvalue to solve for the eigenvectors by setting up and solving a system of equations.
Yes, a 2nd multiplicity eigenvalue can have more than two eigenvectors. The number of eigenvectors associated with an eigenvalue is equal to its multiplicity. So, if a 2nd multiplicity eigenvalue has a multiplicity of 3, it will have three eigenvectors associated with it.