Exact and Numerical Model Contracts
A matrix represents a map only after its input/output spaces and coordinates are specified. A numerical answer needs additional choices: units, precision, norm, tolerances, objective and acceptance criteria.
| Choice | Obligation |
|---|---|
| Field | Core theory uses real scalars; exact labs use rational inputs. Complex adjoints require conjugation. |
| Shape | A is m-by-n, x has n entries, b has m; state input/output bases. |
| Exact versus approximate | Fractions establish exact identities; floating residuals support tolerance-based claims. |
| Rank | Exact rank is algebraic; numerical rank needs a scale, cutoff and perturbation model. |
| Objective | Exact solve, unweighted least squares, weighted least squares and ridge are different problems. |
| Evidence | Solution, impossibility witness, reconstruction, orthogonality, residual and error bound answer different questions. |
Exact systems and coordinates
Row operations must act on both A and b. They preserve the solution set, not the original column space as a set of output vectors. Extract a column-space basis from original pivot columns. A nullspace basis must have n−rank(A) independent vectors, not merely several vectors that happen to map to zero.
For inconsistency, y^T A=0 and y^T b!=0 form a certificate: Ax=b would imply 0=y^T b. For consistency, prove the full set is x_p+Null(A). Rectangular basis matrices have coordinates determined by a solve, not by a square inverse.
Approximation, identifiability and projections
Unweighted least squares minimizes ||Ax−b||2². All minimizers have the same fitted vector, characterized by A^T(b−Ax)=0. Coefficients are unique exactly when the columns are independent. For positive weights W, the corresponding condition is A^T W(b−Ax)=0; transform both A and b by sqrt(W).
Ridge minimizes ||Ax−b||2²+lambda*||x||2² for lambda>0. Its condition is A^T(b−Ax)=lambda*x. It changes the objective and can change the answer even with exact input and arithmetic. State which coefficients are penalized; the small exercises penalize every coefficient, including an intercept if present.
Numerical checks need a scale
Check A≈QR and Q^TQ≈I separately. For Cholesky, check symmetry, positive diagonal in L, and A≈LL^T. A small reconstruction residual does not itself establish accurate coefficients or a reliable rank decision.
The kit reports exact infinity-norm solve diagnostics:
r=b−Ax, eta=||r||/(||A||*||x||+||b||) and, when defined,
relative_forward_error <= kappa(A)*||r||/||b||.
The latter compares x with the exact solution of the supplied nonsingular system,
not with an unknown physical truth. b=0 and zero denominators need explicit
conventions. See the kit contract.
Power-iteration convergence means the returned eigenpair has a sufficiently small residual; it does not certify dominance. SVD truncation has separate operator and Frobenius error formulas. Pseudoinversion after discarding a positive singular value is pseudoinversion of a changed model.
Work and limits
The offline kit caps dimensions at eight and validates its inputs. It is a teaching implementation, not a replacement for a numerical library. No performance claim follows from tiny tests. Explain general invariants separately from finite test evidence, and record any library version used for optional comparisons.