Decomposition theorems of matrices involving idempotents? Is there any non-trivial decomposition of matrices that involve idempotents that provide the shortest sum description of given matrix?
 A: Yes, the spectral decomposition of a symmetric matrix (or a Hermitian matrix). To be more precise, let $A$ be symmetric. Then $$A=\sum_{\lambda\in \sigma(A)}\lambda P_\lambda,$$
where $\sigma(A)$ is the spectrum of $A$, i.e. the eigenvalues of $A$ and $P_{\lambda}$ is the orthogonal projection onto the corresponding eigenspace. Obviously $P_{\lambda}^2=P_{\lambda}$. Hence $P_{\lambda}$ is idempotent.
The above decompostion holds for normal matrices as pointed out in the comments.
Edit: Let's look at the $2\times 2$ matrices. Any idempotent can be written as $\begin{pmatrix}
a& b\\ c&1-a
\end{pmatrix}$ with $a^2+bc=a$. Notice that the trace of such a matrix is $1$. Given any matrix $X=\begin{pmatrix}
x& y\\ z&t
\end{pmatrix}$, is it possible to write $X=\lambda A+\mu B$ where $A$ and $B$ are idempotent? If the answer is no, is it possible to write $X=\sum_{i=1}^n\lambda_iA_i$ with the $A_i$ idempotent? These are interesting first questions to look at. I don't know the answer, and perhaps the non-linear condition $a^2+bc=a$ for idempotent matrices might be very difficult to work with.
