Singular values and the Picard condition

A singular system consists of positive numbers σⱼ and orthonormal vectors vⱼ in X and uⱼ in Y, satisfying Avⱼ=σⱼuⱼ and A*uⱼ=σⱼvⱼ. The v vectors form a basis of the orthogonal complement of the nullspace of A; the u vectors span the closure of its range. They need not form bases of all of X or all of Y. Invisible object components and data orthogonal to the range must therefore be handled separately from the displayed singular coefficients.

Every object has a unique decomposition into an observable component and a null component. Applying A discards the latter and multiplies each observable coefficient by its σⱼ. Exact data in the range must be orthogonal to the nullspace of A*. However, that orthogonality only places data in the closure of the range, which may be larger than the range itself. The distinction is essential: arbitrarily accurate finite approximations to data do not ensure that a finite-norm exact preimage exists.

The range criterion says that y belongs to the range of A exactly when it is orthogonal to N(A*) and the sum of |⟨y,uⱼ⟩|²/σⱼ² is finite. Necessity follows by expanding any solution in the v basis and using Parseval. For sufficiency, that sum makes the proposed inverse series converge in X; boundedness of A permits passage through the limit and reconstructs y using its range-closure expansion. This proves existence rather than merely giving a formal expression.

The resulting solution lies in N(A) perpendicular and has minimum norm. All other exact solutions add a null vector, increasing the squared norm by an orthogonal term. For general data, the Moore–Penrose inverse is defined on R(A) plus its orthogonal complement: it discards the latter and inverts the range component when that component satisfies Picard. It is not defined as a finite-norm object on every datum in Y when the range is not closed. Calling it a universal bounded inverse would contradict compact ill-posedness.

Small data norm does not control the weighted Picard sum because division by small singular values magnifies high-index components. Even a datum supported on one high mode can produce a large inverse. More general perturbations can make the divided coefficients fail square summability altogether. Spectral truncation avoids the infinite tail, but the number of retained modes then becomes a regularization choice. A finite-dimensional solve is not automatically a faithful approximation if that choice ignores the noise level.

The phrase discrete Picard condition often describes observed coefficients decaying faster than singular values over an initial range of indices before a noise plateau. It is a useful diagnostic, not the infinite-dimensional theorem itself. Real data can have correlated errors, and coefficients estimated from an uncertain forward operator may not display a clean plateau. The theorem gives a precise functional-space target; practical diagnostics require additional assumptions about measurement noise and discretization. A visual coefficient plot cannot prove exact membership in a nonclosed range.