## Projection matrices

Out of the unit matrix and a real non-invertible symmetric matrix of the same size,

$$\delta_{ij}$$ and $$M_{ij}$$

I need to build a set of projection matrices, $A_{ij}$ and $B_{ij}$ which satisfy orthonormality:

$$A_{ij} B_{jk}=0,$$ and $$A_{ij} A_{jk}=B_{ij} B_{jk}=\delta_{ik}$$

Is this possible? or should I give up trying to find such matrices?
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 Where does the matrix M come in? I don't think what you're requesting is possible. Just writing in terms of matrices, you want AB = 0 and A2 = B2 = I. But the first condition shows that det(A)det(B) = 0, so det(A) = 0 or det(B) = 0. If det(A) = 0, then det(A2) = 0, making A2 = I impossible.
 Good point; there are no such matrices I can construct. Thanks.