Sorry this question was already asked but my english is not good. For matrix to be similar, does it have to have all of these properties or SOME of them?

Same determinant

Same Trace

Same characteristic polynomial

Same Eigenvalues

P^-1BP = D etc

If one of these properties are not shared by two matrix, does that mean that they are not similar?

Feel free to mentions if I miss some properties

  • $\begingroup$ Well two matrices $B,D$ are defined to be similar if there exists some invertible matrix $P$ such that $P^{-1}BP=D$. It is a theorem that if two matrices are similar they will share the other properties you list. $\endgroup$ – Alex Becker Apr 23 '14 at 20:44

By definition two square matrices $A$ and $B$ are called similar if there's an invertible matrix $P$ such that $$A=PBP^{-1}$$ and all the other conditions are necessary and not sufficient. Here's a counterexample:

$$A=\begin{pmatrix}1&1\\0&1\end{pmatrix}\quad I_2=\begin{pmatrix}1&0\\0&1\end{pmatrix}$$ these two matrices have the same determinant, same eigenvalue, same trace, same charecteristic polynomial but they not equivalent.

  • $\begingroup$ wouldn't it be $A = P^−1BP$ ? $\endgroup$ – Mac Apr 23 '14 at 21:02
  • 1
    $\begingroup$ It's the same: take $Q=P^{-1}$ in your equality. $\endgroup$ – user63181 Apr 23 '14 at 21:09

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