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What is the difference between an "involuted" and an "idempotent" matrix?

I believe that they both have to do with inverse, perhaps "self inverse" matrices.

Or do they happen to refer to the same thing?

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up vote 12 down vote accepted

A matrix $A$ is an involution if it is its own inverse, ie if

$$A^2 = I$$

A matrix $B$ is idempotent if it squares to itself, ie if

$$B^2 = B$$

The only invertible idempotent matrix is the identity matrix, which can be seen by multiplying both sides of the above equation by $B^{-1}$. An idempotent matrix is also known as a projection.

Involutions and idempotents are related to one another. If $A$ is idempotent then $I - 2A$ is an involution, and if $B$ is an involution, then $\tfrac{1}{2}(I\pm B)$ is idempotent.

Finally, if $B$ is idempotent then $I-B$ is also idempotent and if $A$ is an involution then $-A$ is also an involution.

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