Question 15237: 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=[[13, 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=[-1, -1], right vector=[1, 1]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).

• G001 — Use the same real plane with symmetric positive-definite matrix G=[[13, 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=[-1, -1], right vector=[1, 1]. 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(13, 5)

Hilbert Gram diagonal product = 65

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

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(65, 4)

Hilbert Gram determinant = 61

Hilbert completing-square slope := div(2, 13)

Hilbert completing-square slope = 2/13

Hilbert completing-square positive residual := div(61, 13)

Hilbert completing-square positive residual = 61/13

complete real carrier R^2;Gram matrix=[[13, 2], [2, 5]];a=13>0;det=61>0;dimension declaration=2

x^T M x=13*(x0+(2/13)*x1)^2+(61/13)*x1^2;zero only at x=0

Hilbert distance coordinate0 difference := sub(-1, 1)

Hilbert distance coordinate0 difference = -2

Hilbert distance coordinate1 difference := sub(-1, 1)

Hilbert distance coordinate1 difference = -2

Hilbert distance coefficient(0,0) times left coordinate := mul(13, -2)

Hilbert distance coefficient(0,0) times left coordinate = -26

Hilbert distance term(0,0) := mul(-26, -2)

Hilbert distance term(0,0) = 52

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

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

Hilbert distance term(0,1) := mul(-4, -2)

Hilbert distance term(0,1) = 8

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

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

Hilbert distance term(1,0) := mul(-4, -2)

Hilbert distance term(1,0) = 8

Hilbert distance coefficient(1,1) times left coordinate := mul(5, -2)

Hilbert distance coefficient(1,1) times left coordinate = -10

Hilbert distance term(1,1) := mul(-10, -2)

Hilbert distance term(1,1) = 20

Hilbert distance complete bilinear sum := sum(52, 8, 8, 20)

Hilbert distance complete bilinear sum = 88

left=[-1, -1];right=[1, 1];same induced metric on R^2;squared norm=88

distance=sqrt(88)

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

([-1, -1], [1, 1], [-2, -2], 88, sqrt(88))

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.