Question 15231: Compute Distance in the Norm Induced by This Positive-Definite Form

Original question and givens

Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-7, -2], right vector=[7, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).

• G001 — Use the same real plane with symmetric positive-definite matrix G=[[7, 2], [2, 5]] and inner product <x,y>=x^T G y. Its induced norm and metric belong to that same carrier, not to unrelated structures. Left vector=[-7, -2], right vector=[7, 2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).

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-hilbert-space

Step-by-step derivation

Hilbert Gram diagonal product := mul(7, 5)

Hilbert Gram diagonal product = 35

Hilbert Gram off-diagonal product := mul(2, 2)

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(35, 4)

Hilbert Gram determinant = 31

Hilbert completing-square slope := div(2, 7)

Hilbert completing-square slope = 2/7

Hilbert completing-square positive residual := div(31, 7)

Hilbert completing-square positive residual = 31/7

complete real carrier R^2;Gram matrix=[[7, 2], [2, 5]];a=7>0;det=31>0;dimension declaration=2

x^T M x=7*(x0+(2/7)*x1)^2+(31/7)*x1^2;zero only at x=0

Hilbert distance coordinate0 difference := sub(-7, 7)

Hilbert distance coordinate0 difference = -14

Hilbert distance coordinate1 difference := sub(-2, 2)

Hilbert distance coordinate1 difference = -4

Hilbert distance coefficient(0,0) times left coordinate := mul(7, -14)

Hilbert distance coefficient(0,0) times left coordinate = -98

Hilbert distance term(0,0) := mul(-98, -14)

Hilbert distance term(0,0) = 1372

Hilbert distance coefficient(0,1) times left coordinate := mul(2, -14)

Hilbert distance coefficient(0,1) times left coordinate = -28

Hilbert distance term(0,1) := mul(-28, -4)

Hilbert distance term(0,1) = 112

Hilbert distance coefficient(1,0) times left coordinate := mul(2, -4)

Hilbert distance coefficient(1,0) times left coordinate = -8

Hilbert distance term(1,0) := mul(-8, -14)

Hilbert distance term(1,0) = 112

Hilbert distance coefficient(1,1) times left coordinate := mul(5, -4)

Hilbert distance coefficient(1,1) times left coordinate = -20

Hilbert distance term(1,1) := mul(-20, -4)

Hilbert distance term(1,1) = 80

Hilbert distance complete bilinear sum := sum(1372, 112, 112, 80)

Hilbert distance complete bilinear sum = 1676

left=[-7, -2];right=[7, 2];same induced metric on R^2;squared norm=1676

distance=sqrt(1676)

Reverse checks

• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute Hilbert distance complete bilinear sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Hilbert distance term(1,1) with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Hilbert distance coefficient(1,1) times left coordinate with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Hilbert distance term(1,0) with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Hilbert distance coefficient(1,0) times left coordinate with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Hilbert distance term(0,1) with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Hilbert distance coefficient(0,1) times left coordinate with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Hilbert distance term(0,0) with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Hilbert distance coefficient(0,0) times left coordinate with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Hilbert distance coordinate1 difference with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Hilbert distance coordinate0 difference with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute Hilbert completing-square positive residual with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Hilbert completing-square slope with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Hilbert Gram determinant with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Hilbert Gram off-diagonal product with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Recompute Hilbert Gram diagonal product with exact rational arithmetic, not rounded decimals.

Result

([-7, -2], [7, 2], [-14, -4], 1676, sqrt(1676))

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.