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

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