A 3x3 matrix A can indeed have 0 as an eigenvalue, but this occurs only when the matrix is singular, meaning it is not invertible. The discussion highlights that if 0 is an eigenvalue, then the equation Mx = 0 must have a nontrivial solution, which implies the existence of a non-zero vector x. Therefore, any matrix with a column or row of zeros will have 0 as an eigenvalue, while invertible matrices cannot have 0 as an eigenvalue.
PREREQUISITESStudents of linear algebra, mathematicians, and anyone interested in understanding the properties of matrices and eigenvalues.
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?