# Facts About Eigenvectors and Eigenvalues

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:

1. If Ax=λx for some vector x, then λ is an eigenvalue of A.

2. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy.

3. A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution x.
4. A matrix A is not invertible if and only if 0 is an eigenvalue of A.
5. To find the eigenvalues of A, reduce A to echelon form.
6. If Ax=λx for some vector x, then x is an eigenvector of A.
7. An eigenspace of A is just a null space of a certain matrix.
8. The eigenvalues of a matrix are on its main diagonal.
9. A steady-state vector for a stochastic matrix is actually an eigenvector.
10. 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.

• 1 is false, since $x$ might be zero. Jan 30 '20 at 19:12
• One question per post, please. Jan 30 '20 at 21:21

1. False, because $$x$$ might be zero.

2. True. Finding eigenvectors requires solving a linear system, while the verification just requires a matrix-vector multiplication.

3. True, this is essentially the definition of an eigenvalue.

4. False, $$A$$ is invertible if and only if 0 is not an eigenvalue of $$A$$.

5. False, row operations can change the eigenvalues of a matrix.

6. False. Again, $$x$$ might be zero.

7. True, it's the null space of $$A - \lambda I$$, where $$\lambda$$ is an eigenvalue.

8. False, this holds for triangular matrices, but not in general.

9. True. A steady-state satisfies $$Ax = x$$, so $$x$$ is an eigenvector corresponding to $$\lambda = 1$$.

10. False. Consider, for example, the identity matrix. Both $$(0,1)$$ and $$(1,0)$$ are eigenvectors corresponding to $$\lambda = 1$$.

• Thank you so much! Jan 31 '20 at 17:45