Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [-2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
G001 — Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-4, -2], [-2, 4]]. All input vector dimensions must match the matrix. Distributivity proves linearity separately in both slots for all real vectors and scalars; individual integer evaluations illustrate that fixed formula, not a black-box sampling proof. Symmetry, positive definiteness, nondegeneracy and inner-product qualification are additional conditions. Give (matrix,first [i,j,A_ij,A_ji] mismatch with i<j in row order or null,symmetric?,symmetry_is_extra_not_bilinearity).
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
fixed real matrix=[[-4, -2], [-2, 4]];B(x,y)=sum_i sum_j x_i*A_ij*y_j
After
distributivity proves real linearity in each slot separately for all vectors and scalars
Conditions: none
Distributivity expands any real linear combination in either slot of the finite double sum,giving separate universal linearity certificates.Integer instances evaluate this fixed formula,not prove a black box by sampling.
Standard basis vectors give B(e_i,e_j)=A_ij,so a mismatch immediately refutes symmetry.All opposite entries must agree for the universal A=A^T certificate.
If A=A^T,exchanging indices in the double sum gives symmetry for all real vectors.Symmetry still does not imply positive definiteness,nondegeneracy or an inner product;the zero matrix is an example.
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.