Question 13879: Symmetry Alone Does Not Establish an Inner Product

Original question and givens

Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

• G001 — Use the exact real R² candidate B(x,y)=x^T A y with the complete integer matrix A=[[5, 2], [2, 5]]. Bilinearity follows from the formula; symmetry and strict positive definiteness remain separate obligations. For a symmetric 2-by-2 matrix the Sylvester certificate requires a positive leading entry and positive determinant. Positive semidefiniteness or finitely many positive samples are insufficient, and completeness is not inferred. Give (matrix,[A_01,A_10],symmetric?).

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.

• mko-real-inner-product-space

Step-by-step derivation

candidate real R2 form;matrix=[[5, 2], [2, 5]];B(x,y)=x^T*A*y

matrix distributivity supplies bilinearity;other inner-product obligations remain separate

opposite off-diagonal entries=[2, 2]

symmetric=Yes

Reverse checks

• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[5, 2], [2, 5]], [2, 2], Yes)

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.