Is X Always Commutative with Y in Matrix Invariance?

  • MHB
  • Thread starter Euge
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    2015
In summary, to access this week's POTW, you can visit the official website of the organization or publication that hosts it. The POTW is a weekly challenge designed to test your problem-solving and critical thinking skills. Each organization may have their own restrictions and guidelines for solving the POTW, so it is important to check before working with others. Solving the POTW can improve your skills and may offer rewards or recognition from the hosting organization.
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Euge
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Since many schools have just started the school year, I'd like to begin September's POTW with an important, yet simple problem.

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Suppose $X$ and $Y$ are $n \times n$ complex matrices such that $X$ is $Y$-invariant, i.e., $e^{zY}Xe^{-zY} = X$ for all $z\in \Bbb C$. Prove that $X$ and $Y$ commute.
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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
No one solved this week's problem correctly. You can read my solution below.
Differentiate the equation $e^{zY}Xe^{-zY} = X$ with respect to $z$ to obtain

$$Ye^{zY}Xe^{-zY} - e^{zY}XYe^{-zY} = 0,$$

then set $z = 0$ to get $YX - XY = 0$, or $YX = XY$.
 

FAQ: Is X Always Commutative with Y in Matrix Invariance?

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