Question 15234: 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=[[10, 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=[-4, 1], right vector=[4, -2]. Give (left,right,left_minus_right,squared_distance,symbolic_square_root).

• G001 — Use the same real plane with symmetric positive-definite matrix G=[[10, 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=[-4, 1], right vector=[4, -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(10, 5)

Hilbert Gram diagonal product = 50

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

Hilbert Gram off-diagonal product = 4

Hilbert Gram determinant := sub(50, 4)

Hilbert Gram determinant = 46

Hilbert completing-square slope := div(2, 10)

Hilbert completing-square slope = 1/5

Hilbert completing-square positive residual := div(46, 10)

Hilbert completing-square positive residual = 23/5

complete real carrier R^2;Gram matrix=[[10, 2], [2, 5]];a=10>0;det=46>0;dimension declaration=2

x^T M x=10*(x0+(1/5)*x1)^2+(23/5)*x1^2;zero only at x=0

Hilbert distance coordinate0 difference := sub(-4, 4)

Hilbert distance coordinate0 difference = -8

Hilbert distance coordinate1 difference := sub(1, -2)

Hilbert distance coordinate1 difference = 3

Hilbert distance coefficient(0,0) times left coordinate := mul(10, -8)

Hilbert distance coefficient(0,0) times left coordinate = -80

Hilbert distance term(0,0) := mul(-80, -8)

Hilbert distance term(0,0) = 640

Hilbert distance coefficient(0,1) times left coordinate := mul(2, -8)

Hilbert distance coefficient(0,1) times left coordinate = -16

Hilbert distance term(0,1) := mul(-16, 3)

Hilbert distance term(0,1) = -48

Hilbert distance coefficient(1,0) times left coordinate := mul(2, 3)

Hilbert distance coefficient(1,0) times left coordinate = 6

Hilbert distance term(1,0) := mul(6, -8)

Hilbert distance term(1,0) = -48

Hilbert distance coefficient(1,1) times left coordinate := mul(5, 3)

Hilbert distance coefficient(1,1) times left coordinate = 15

Hilbert distance term(1,1) := mul(15, 3)

Hilbert distance term(1,1) = 45

Hilbert distance complete bilinear sum := sum(640, -48, -48, 45)

Hilbert distance complete bilinear sum = 589

left=[-4, 1];right=[4, -2];same induced metric on R^2;squared norm=589

distance=sqrt(589)

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

([-4, 1], [4, -2], [-8, 3], 589, sqrt(589))

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.