Projection matrices

TriTertButoxy

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

[tex]\delta_{ij}[/tex] and [tex]M_{ij}[/tex]

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

[tex]A_{ij} B_{jk}=0,[/tex] and [tex]A_{ij} A_{jk}=B_{ij} B_{jk}=\delta_{ik}[/tex]

Is this possible? or should I give up trying to find such matrices?

adriank

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.

TriTertButoxy

Good point; there are no such matrices I can construct. Thanks.

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adriank	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.

TriTertButoxy	Good point; there are no such matrices I can construct. Thanks.