Can An Idempotent Matrix Be Complex?


Answer :

A assume that by "can AA be complex", you mean "can AA have any non-real entries". Well, it can! For instance, take A = \pmatrix{1&i\\0&0} In general: for any complex column-vector xx, A=xx∗x∗xA = \frac{xx^*}{x^*x} (where ∗* denotes the conjugate-transpose) is such a matrix.


A projection to a subspace is idempotent. Therefore AA has no reason to be real. For example, take a subspace SS of C2\mathbb{C}^2 and AA be the matrix of the projection on to SS with respect to the standard basis.


Any matrix A = \pmatrix{a&b\\c&1-a} will be idempotent provided that a2+bc=aa^2+bc=a


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