In a lot of my quantum mechanics and linear algebra books, operators are often defined as $M=IMI$ and many operations on operators are often done in similar fashion, for eg. operator $M$ might be sandwhiched between $H$ and $H^+$ (hermitian conjugate of H). What does $M=IMI$ mean and why is it written like that? Here is another example.

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    $\begingroup$ That is not a definition of $M$. It is simply an equaton that always holds. (For those that can't be bothered to click on the link: In the linked example, $I$ is expressed as $P+Q$, and then $M=(P+Q)M(P+Q)$ is expanded and clever things are deduced.) $\endgroup$ – TonyK Aug 13 at 10:12
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    $\begingroup$ In a nutshell: if $H$ is invertible, then $HMH^{-1}$ is a similar matrix to $M$ and $HMH^\dagger$ is a congruent matrix to $M$. If $H$ is unitary, then we can have both. Matrix similarity comes out of studying linear transformations under changes of basis, and matrix congruence comes out of studying bilinear forms under changes of basis. $\endgroup$ – Omnomnomnom Aug 13 at 10:33

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