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:
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If Ax=λx for some vector x, then λ is an eigenvalue of A.
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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.
Best Answer
False, because $x$ might be zero.
True. Finding eigenvectors requires solving a linear system, while the verification just requires a matrix-vector multiplication.
True, this is essentially the definition of an eigenvalue.
False, $A$ is invertible if and only if 0 is not an eigenvalue of $A$.
False, row operations can change the eigenvalues of a matrix.
False. Again, $x$ might be zero.
True, it's the null space of $A - \lambda I$, where $\lambda$ is an eigenvalue.
False, this holds for triangular matrices, but not in general.
True. A steady-state satisfies $Ax = x$, so $x$ is an eigenvector corresponding to $\lambda = 1$.
False. Consider, for example, the identity matrix. Both $(0,1)$ and $(1,0)$ are eigenvectors corresponding to $\lambda = 1$.