# Question 12778: Symmetry Is Additional to Bilinearity

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0024-12778 · q-20261014-advance-20261027-r04-real-bilinear-form-0078



## Original question and givens

Use B(x,y)=x^T A y over the real coordinate space with square integer matrix A=[[-10, -1], [5, -2]]. 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=[[-10, -1], [5, -2]]. 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.


- [mko-real-bilinear-form](https://ocme.evemisslab.com/en/mko/mko-real-bilinear-form/)





## Step-by-step derivation

S001 · impliesBefore

fixed real matrix=[[-10, -1], [5, -2]];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.

Step identifiers and dependenciesRule: matrix_formula_bilinearity_universal_not_samples · Uses: G001





S002 · impliesBefore

matrix coordinates=[0, 1];A_ij=-1;A_ji=5

After

opposite entries equal=No

Conditions: none

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.

Step identifiers and dependenciesRule: bilinear_each_opposite_matrix_entry · Uses: G001, S001





S003 · impliesBefore

first mismatch in row order=[0, 1, -1, 5]

After

symmetric=No;symmetry_is_extra_not_bilinearity

Conditions: none

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.

Step identifiers and dependenciesRule: bilinear_symmetry_additional_not_inner_product · Uses: G001, S002









## Reverse checks


- S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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

([[-10, -1], [5, -2]], [0, 1, -1, 5], No, symmetry_is_extra_not_bilinearity)

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 50d92cc217149e6a9e2b11cc646ff86ec21a8e908d71f56e157764ec4d8c8563
- candidate record SHA-256: 706bb3edf00cdc594f265f7f0cb1de53afc0a5f1fd28e8c74b56a5741b19a2c4
- semantic review receipt SHA-256: 1c8f6c2c6c75852cb93c08235684ce9f32ebe86065c58c693a04257e772c2d35
- question-first witness SHA-256: 05fd764b5f9f7fff6185003efe576d87632f6daf40606ac3839a7feafe4f80d8
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261027-r04-real-bilinear-form-0078.json)