Some questions about eigenvalues, eigenvectors, and diagonalization

Question

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.

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.