I have to show that the following matrices are not similar:
$$A = \left[\begin{matrix} 1 & 3 & -3 \\ -3 & 7 & -3 \\ -6 & 6 & -2\end{matrix}\right]$$
and $$A' = \left[\begin{matrix} 5 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 3\end{matrix}\right]$$
I know that 2 matrices $A$ and $A'$ are similar if there exists an invertible matrix $B$, such that $$A'= B^{-1}AB$$
According to Wikipedia, similar matrices share some properties (for example they have the same eigen values), but I don't know how to start, since I have missed the last lectures of my linear algebra course, unfortunetely. Unfortuntely, life has not been so completely fair with me 🙁
Should I just check if they have the same eigen values?
Best Answer
Since adding numbers along the diagonal is simple, you can start with the fact that the trace is preserved by similarity transformations, i.e. $\text{tr}(A) = \text{tr}(B^{-1}AB)$.
Do $A$ and $A'$ have the same trace? If not, then $A$ and $A'$ are not similar.
EDIT: The trace of a square matrix is simply the sum of the entries on the diagonal, i.e. for an $n \times n$ matrix $A$, we have $\text{tr}(A) = \displaystyle\sum_{i = 1}^{n}A_{i,i}$.