Can An Idempotent Matrix Be Complex?
Answer : A assume that by "can A A A be complex", you mean "can A A A have any non-real entries". Well, it can! For instance, take A = \pmatrix{1&i\\0&0} In general: for any complex column-vector x x x , A = x x ∗ x ∗ x A = \frac{xx^*}{x^*x} A = x ∗ x x x ∗ (where ∗ * ∗ denotes the conjugate-transpose) is such a matrix. A projection to a subspace is idempotent. Therefore A A A has no reason to be real. For example, take a subspace S S S of C 2 \mathbb{C}^2 C 2 and A A A be the matrix of the projection on to S S S with respect to the standard basis. Any matrix A = \pmatrix{a&b\\c&1-a} will be idempotent provided that a 2 + b c = a a^2+bc=a a 2 + b c = a