I am currently studying Eigenvalues and Eigenvectors in college, and I am supposed to mark true or false on a list of facts about them. I have done so, but it is saying that I am incorrect on the ones that I have marked; I believe that the quiz may be broken, so I was hoping to get some clarification. The facts are:
If Ax=λx for some vector x, then λ is an eigenvalue of A.
Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy.
- A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution x.
- A matrix A is not invertible if and only if 0 is an eigenvalue of A.
- To find the eigenvalues of A, reduce A to echelon form.
- If Ax=λx for some vector x, then x is an eigenvector of A.
- An eigenspace of A is just a null space of a certain matrix.
- The eigenvalues of a matrix are on its main diagonal.
- A steady-state vector for a stochastic matrix is actually an eigenvector.
- If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
I believe that the facts that are true are 1, 3, 4, 5, 6, 7, 8, 9. Would someone be able to clarify which are true and false? Thank you.