In linear algebra, idempotent matrices are defined by
$$
A^2 = A \tag{1}
$$
for a square matrix $A$. Obviously, the identity matrix $I$ is an idempotent matrix. It can be also shown that if $M$ is idempotent, then $I – M$ is idempotent by a trivial calculation.
$$
(I – M) (I – M) = I – M – M + M^2 = I – M – M + M = I – M
$$
In a similar manner, we can define an anti-idempotent matrix $A$ by the condition
$$
A^2 = – A \tag{2}
$$
(A trivial example is the zero matrix.)
To find non-trivial examples of an anti-idempotent matrix $A$, I considered the case of $(2 \times 2)$ matrices:
$$
A = \left[ \matrix{ a & b \cr
c & d \cr} \right]
$$
If $A$ is anti-idempotent, then it must satisfy: $A^2 = -A$.
This leads to a set of $4$ equations:
$$
a^2 + b c = – a, \ a b + b d = – b, \ c a + c d = -c, \ \ b c + d^2 = -d
$$
A simple manipulation results in the equations
$$
b (a + d + 1) = 0, c (a + d + 1) = 0, a^2 + b c = -a, b c + d^2 = -d.
$$
Taking $a = 2$, we see that $a + d + 1 = 0$ or $d = -3$.
We can choose $b$ and $c$ from $b c = -6$. One choice is $b = 2, c = -3$.
Thus, an anti-idempotent matrix is:
$$
A = \left[ \begin{array}{cc}
2 & 2 \\
-3 & -3 \\
\end{array} \right]
$$
(It is easy to check that $A^2 = -A$.)
It can be easily shown : If $M$ is an anti-idempotent matrix, then $I + M$ is also anti-idempotent. Indeed,
$$
(I + M) (I + M) = I + M + M + M^2 = I + M + M – M = I + M.
$$
The examples I considered for anti-idempotent matrices yield singular matrices.
I like to know if it is generally true that anti-idempotent matrices are singular matrices. How to establish this result? Your comments are welcome.
Best Answer
They are just matrices that satisfy $p(A) = 0$ where $p(z) = z^2 + z$. Note that $p$ is square-free, and hence any such $A$ is diagonalisable. The eigenvalues of $A$ are restricted to the roots of $p$, i.e. consisting of $0$ or $-1$, and nothing else.
So, the invertible anti-idempotent matrix are diagonalisable and have only $-1$ as an eigenvalue. It's not hard to see that $-I$ is the only possibility. Every other anti-idempotent matrix will not be invertible.
Another thing to note: $A$ is anti-idempotent if and only if $-A$ is idempotent!