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?
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?
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.