Give examples of two invertible linear operators $T$ and $U$ such that $TU=-UT$ holds.
Please help me in finding this example.
Thank you in advance.
Problem in finding examples of linear 0perators $T$ and $U$ such that $TU=-UT$ holds.
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linear-algebra
linear-transformations
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1To begin, what does $TU=-UT$ tell you about the eigenvalues of $T$ and $U$? – 2017-02-18
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0**Hint:** For invertible $U$, this becomes $$UTU^{-1}=-T$$ – 2017-02-18
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0But can you provide me an example? – 2017-02-18
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1@Arnab of course. However, we'd prefer that you put some amount of effort into answering the question yourself first. – 2017-02-18
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0For Googling purposes, such operators are said to _anticommute_. – 2017-02-18
1 Answers
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For example, take $$ \pmatrix{0&1\\1&0}, \qquad \pmatrix{1&0\\0&-1} $$
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0If one is willing to work with complex Hermitean matrices, then one can include $\begin{pmatrix}0&i\\-i&0\end{pmatrix}$ which anticommutes with the above as well. (Together, these are known in physics as the three Pauli matrices and show up frequently in quantum mechanics.) – 2017-02-18
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0I have found one such.Which is $\pmatrix {0&1\\-1&0}$ and $\pmatrix {0&-1\\-1&0}$.Also I found that for such matrix $P$ and $Q$, $Trace(P) = Trace(Q) = 0$.Sorry for the delay to respond.The reason is the time when I asked it was a time for me to sleep. – 2017-02-19
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0No problem. Glad you figured it out! The point of my comment was to note that $T$ is similar to $-T$, just as $U$ is similar to $-U$. – 2017-02-19