1.5 Selection, residuals, and information loss
On this page · Permalink · Source line 457
For any set \(S\subseteq I_N\), let \(D_S\) be its diagonal indicator and \(P_S=F_ND_SF_N^\dagger\). This construction does not require \(S\) to be an order class. Rationality is special, but orthogonal projection is general.
Theorem QA07: Projector geometry and exact residual reconstruction
For every \(x\in\mathbb C^N\),
Furthermore, \(P_S\) is injective exactly when \(S=I_N\). If \(S\) is proper, its kernel has dimension \(N-|S|\).
Proof
The first equality is distributivity. Since \(P_S^\dagger=P_S\) and \(P_S^2=P_S\),
Expanding the norm square of the orthogonal sum proves the energy identity. In Fourier coordinates, \(P_S\) is \(D_S\), whose kernel consists exactly of the vectors supported outside \(S\). Its dimension is \(N-|S|\). This kernel is zero exactly when every label is selected. \(\square\)
This gives a precise reconstruction contract. The pair of packets \((P_Sx,(I-P_S)x)\) retains the complete vector, subject to the compatibility condition that the first lies in the range and the second in the kernel. Keeping only the first packet is an exact projection but an incomplete representation of an arbitrary input.
For character-order packets, QA06 gives the stronger complete decomposition
The decomposition is a change in organization rather than a reduction in the dimension of the whole signal. Smaller stored selectors do not make an arbitrary signal contain less information. Compression requires a restriction on the admissible signals or an explicit approximation criterion.