Question 13953: The Exact l1 Operator Norm Uses Absolute Column Sums

Original question and givens

Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.

• G001 — Use the explicit rectangular integer matrix A=[[-10, -1, 1], [0, 10, 4]] as a linear map between its column- and row-coordinate real spaces. Both ends use the same stated l1 or linfinity norm. A single C≥0 must bound every input: norm(A*x)≤C*norm(x). The exact l1 induced norm is the maximum absolute column sum; the exact linfinity induced norm is the maximum absolute row sum. An instance pass or a nonoptimal usable bound is not the exact operator norm. Give (matrix,column_sums,operator_norm,maximum_absolute_column_sum), preserving original column order.

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-bounded-linear-operator

Step-by-step derivation

original column index=0;ordered entries=[-10, 0]

absolute entries=[10, 0]

operator column0 absolute sum := sum(10, 0)

operator column0 absolute sum = 10

original column index=1;ordered entries=[-1, 10]

absolute entries=[1, 10]

operator column1 absolute sum := sum(1, 10)

operator column1 absolute sum = 11

original column index=2;ordered entries=[1, 4]

absolute entries=[1, 4]

operator column2 absolute sum := sum(1, 4)

operator column2 absolute sum = 5

all absolute column sums=[10, 11, 5]

maximum=11;first maximizing column=1

arbitrary x in R^3;all column sums<=11

norm1(Ax)<=sum_j(sum_i abs(Aij))*abs(xj)<=11*norm1(x)

attaining unit input image:matrix=[[-10, -1, 1], [0, 10, 4]];input=[0, 1, 0]

input dimension=3;output dimension=2;rows dot input

attaining unit input image row0 column0 product := mul(-10, 0)

attaining unit input image row0 column0 product = 0

attaining unit input image row0 column1 product := mul(-1, 1)

attaining unit input image row0 column1 product = -1

attaining unit input image row0 column2 product := mul(1, 0)

attaining unit input image row0 column2 product = 0

attaining unit input image row0 sum := sum(0, -1, 0)

attaining unit input image row0 sum = -1

attaining unit input image row1 column0 product := mul(0, 0)

attaining unit input image row1 column0 product = 0

attaining unit input image row1 column1 product := mul(10, 1)

attaining unit input image row1 column1 product = 10

attaining unit input image row1 column2 product := mul(4, 0)

attaining unit input image row1 column2 product = 0

attaining unit input image row1 sum := sum(0, 10, 0)

attaining unit input image row1 sum = 10

attaining input:signed vector=[0, 1, 0];norm=l1

coordinate absolute values=[0, 1, 0]

attaining input l1 sum := sum(0, 1, 0)

attaining input l1 sum = 1

attaining output:signed vector=[-1, 10];norm=l1

coordinate absolute values=[1, 10]

attaining output l1 sum := sum(1, 10)

attaining output l1 sum = 11

explicit input=[0, 1, 0];input norm=1;image=[-1, 10];output norm=11

uniform upper bound=11 and attained ratio=11;exact induced norm=11

Reverse checks

• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Recompute attaining output l1 sum with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute attaining input l1 sum with exact rational arithmetic, not rounded decimals.
• 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 attaining unit input image row1 sum with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute attaining unit input image row1 column2 product with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute attaining unit input image row1 column1 product with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute attaining unit input image row1 column0 product with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute attaining unit input image row0 sum with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute attaining unit input image row0 column2 product with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute attaining unit input image row0 column1 product with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute attaining unit input image row0 column0 product with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute operator column2 absolute sum with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute operator column1 absolute sum with exact rational arithmetic, not rounded decimals.
• 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 — Recompute operator column0 absolute sum with exact rational arithmetic, not rounded decimals.
• 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, 1], [0, 10, 4]], [10, 11, 5], 11, maximum_absolute_column_sum)

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.