0

How do you show that two matrices are similar? I know that if they are similar, then B=(v^-1)AV, but how do you find V^-1 and V? For example, the matrices {[1,2]T, [3,2]T} and {[0,4]T, [2,2]T}, how would you find the P and P^-1 (if they are similar)? Or would you have to show they are similar in a different way?

Thank you!

redlobsterdrummer
  • 75

1 Answers1

0

If you meant the matrices

$$\begin{pmatrix}1&2\\3&2\end{pmatrix}\;,\;\;\;\begin{pmatrix}0&4\\2&2\end{pmatrix}$$

then they can't be similar since their traces are different: $\;3\;,\;\;2\;$ , respectively.

DonAntonio
  • 211,718
  • 17
  • 136
  • 287
  • Yes those are what I mean! So if they have a different trace they can't be similar? If they were similar, how would you find the P and P^-1 though? Sorry still stuck on that concept. But also thank you! – redlobsterdrummer May 08 '16 at 06:50
  • Trace is preserved under similar matrices (or equivalently, the trace of a linear map is preserved under a change of base), this follows from the tracial property $\text{Tr}(AB)=\text{Tr}(BA)$. – Mathematician 42 May 08 '16 at 06:53
  • I might specify, since involving trace might seem weird. But as mentioned, if $A = PBP^{-1}$, then $$\operatorname{Tr}(A) = \operatorname{Tr}((PB)(P^{-1})) = \operatorname{Tr}((P^{-1})(PB)) = \operatorname{Tr}(B)$$. Arguing from the contrapositive we have that $A$ and $B$ are not similar if the trace is not equal. –  May 08 '16 at 06:57
  • @redlobsterdrummer There are several matrix-invariants under similarity, say: same characteristic and minimal polynomials,. same eigenvectors, same rank, same determinant, same trace...If one of these fails then the matrices cannot be similar. – DonAntonio May 08 '16 at 07:02