Is the following equation true about trace of a Matrix?

Asked Jun 16 '16 at 01:14

Active Jun 16 '16 at 03:03

Viewed 67 times

Is this true? $\operatorname{tr}(AB)^k = \operatorname{tr}(A^k B^k)$

If so, how can one provide proof or counter example?

I tried it with the following two matrices and it turned out to be true: \begin{bmatrix}1&0\\0&1\end{bmatrix} \begin{bmatrix}2&0\\0&2\end{bmatrix} but would it apply to all matrices? and how can one provide proof?

edited Jun 16 '16 at 03:03

Michael Hardy

asked Jun 16 '16 at 01:14

user3015986

3

Try some matrices that are less special than the two that you posted before you edited the question. – Ethan Bolker Jun 16 '16 at 01:16
confirmed it with a more normal matrix, and the equation still holds true. – user3015986 Jun 16 '16 at 01:21
1

If by $tr(AB)^k$ you mean $(tr(AB))^k$, this isn't even true with $B=I$. Consider $A=\begin{bmatrix} 3&0\0&3\end{bmatrix}$ and $k=2$. $(tr(AB))^k = 6^2=36$ whereas $tr(A^kB^k)=18$ – JMoravitz Jun 16 '16 at 01:29
If you just generated 2 random matrices (e.g. with entries i.i.d. standard normal; in matlab, randn(n) for n x n), you'd probably find a counterexample on the first try. – Batman Jun 16 '16 at 01:34
See this question for a class of counterexamples – Ben Grossmann Jun 16 '16 at 02:03

Is the following equation true about trace of a Matrix?

0 Answers0