I've been studying for my linear algebra final and was going through the review sheet the professor gave us. Most of the content was easy to understand, but I couldn't get my head around a few concepts. I've listed the questions below:

  1. Provide an example of a 2x2 matrix with one linearly independent eigenvector
  2. Provide an example of matrices which have the same eigenvalues but are not similar.
  3. Provide an example of a matrix with eigenvalues 3 and 4, occurring at multiplicities 5 and 6, respectively.
  4. Suppose that A and B are n × n matrices, and that A is similar to B. Show that A and B have the same characteristic polynomial.

I think I've sort of figured out 4, so I'll show my working:

If $A = PBP^{-1}$, then to show that A and B have the same characteristic polynomials we must find $det (A-\lambda I)$ and prove that it is equal to $det (B-\lambda I)$

$\to$ $det (A - \lambda I) $ = $det (PBP^{-1} - \lambda I) $

= $det (P^{-1}(B- \lambda I)P)$

= $(det P^{-1})(det B- \lambda I)(det P)$

= $ (det B- \lambda I)$

Since the determinants of A and B are equal, we know that their characteristic polynomials will be the same.

I hope this is a coherent explanation; I'm a freshman in college so my exposure to rigorous, inductive mathematics isn't very high.

  • $\begingroup$ Hi! Welcome to MSE. Your answer to 4 is indeed correct, but I would explain a bit more (in particular, I would write down that the identity matrix commutes with all matrices and how you're using that to factor the change of basis matrices). $\endgroup$
    – Otomeram
    Nov 27, 2020 at 7:18
  • 1
    $\begingroup$ @Otomeram ok, thank you for your answer! $\endgroup$ Nov 27, 2020 at 7:27

1 Answer 1

  1. The standard example is $\begin{pmatrix}1&1\\0&1\end{pmatrix}$.

  2. Looking at the matrix above. can you think of another matrix with the same eigenvalues that is not similar to it?

  3. Use a diagonal matrix with enough $3$'s and $4$'s on the diagonal.

  • $\begingroup$ Thank you for your answer! Unfortunately, I can't think of such a matrix. I have a feeling the answer may just be right in front of my eyes but I'm completely missing it. I know that similar matrices have the same determinant, eigenvalues, rank, and characteristic polynomials, so I tried thinking of such a matrix that could contradict one of the properties (obviously not eigenvalues). $\endgroup$ Nov 27, 2020 at 7:41
  • $\begingroup$ What is the simplest $2 \times 2$ matrix whose only eigenvalue is $1$? $\endgroup$
    – KCd
    Nov 27, 2020 at 8:01
  • $\begingroup$ The identity matrix I reckon. $\endgroup$ Nov 27, 2020 at 8:49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.