The discussion revolves around the question of whether a 3x3 matrix can have 0 as an eigenvalue, with participants exploring the implications of matrix invertibility and eigenvalue definitions.
The conversation has highlighted the importance of matrix invertibility in relation to eigenvalues, with some participants questioning the assumptions made about specific matrix types and their eigenvalues. Guidance has been offered regarding the definitions involved.
One participant notes that the matrix must be invertible, which rules out certain examples like the zero matrix or matrices with a column of zeros. This constraint has shifted the focus of the discussion.
D H said:All you need is one example of a 3x3 matrix with zero as an eigenvalue. Hint: Finding one is a trivial task.
jbunniii said:Do you know any types of matrices for which you can immediately determine all the eigenvalues by inspection?
nicknaq said:The identity matrix, the zero matrix and triangular matrices I suppose?
jbunniii said:OK, let's look at the first two. What are the eigenvalues of those matrices?
Could be. Can you come up with a specific example and check it?nicknaq said:Any 3x3 matrix with a column of zeros? A homogeneous system?
nicknaq said:WOOPS!
I forgot one vital piece of information guys! I'm sorry.
The matrix must be invertible.
So that rules out the zero matrix or any matrix with a column/row of zeros.
jbunniii said:That changes everything!
To see if such a matrix can exist, consider the definition: \lambda is an eigenvalue of M if there is a nonzero vector x such that
Mx = \lambda x
If \lambda = 0 is an eigenvalue of M, what does that imply?
nicknaq said:Mx=0 so there's only the trivial solution. However x cannot be zero so there's no solution? Is this my proof or is there more to it?