TITLE: Can an idempotent matrix be complex?
QUESTION [16 upvotes]: A matrix $A$ is called idempotent if $A^2 = A$. I am just wondering if such matrix can be complex. Anyone can help give an example or proof that it has to be real? Thanks!

REPLY [13 votes]: Any matrix $A = \pmatrix{a&b\\c&1-a}$ will be idempotent provided that $a^2+bc=a$