# Question 13972: The Exact l1 Operator Norm Uses Absolute Column Sums [Return to catalogue page 70](https://ocme.evemisslab.com/en/questions/page/70/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261028-r05-bounded-linear-operator-foundations/#q-20261014-advance-20261028-r05-bounded-linear-operator-0022) 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. R0026-13972 · q-20261014-advance-20261028-r05-bounded-linear-operator-0022 ## Original question and givens Use the explicit rectangular integer matrix A=[[9, -3, 3], [-1, -9, -5]] 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=[[9, -3, 3], [-1, -9, -5]] 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](https://ocme.evemisslab.com/en/mko/mko-bounded-linear-operator/) ## Step-by-step derivation S001 · impliesBefore original column index=0;ordered entries=[9, -1] After absolute entries=[9, 1] Conditions: none For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles. Step identifiers and dependenciesRule: operator_correct_axis_absolute_sum · Uses: G001 S002 · calculationBefore operator column0 absolute sum := sum(9, 1) After operator column0 absolute sum = 10 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · impliesBefore original column index=1;ordered entries=[-3, -9] After absolute entries=[3, 9] Conditions: none For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles. Step identifiers and dependenciesRule: operator_correct_axis_absolute_sum · Uses: G001, S002 S004 · calculationBefore operator column1 absolute sum := sum(3, 9) After operator column1 absolute sum = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · impliesBefore original column index=2;ordered entries=[3, -5] After absolute entries=[3, 5] Conditions: none For l1 sum down each column;for linfinity sum across each row.Preserve axis order and do not transpose rectangular roles. Step identifiers and dependenciesRule: operator_correct_axis_absolute_sum · Uses: G001, S004 S006 · calculationBefore operator column2 absolute sum := sum(3, 5) After operator column2 absolute sum = 8 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore all absolute column sums=[10, 12, 8] After maximum=12;first maximizing column=1 Conditions: none Compute every required axis sum before taking a maximum.The value must have both a universal upper bound and an attaining unit input. Step identifiers and dependenciesRule: operator_maximum_axis_sum · Uses: G001, S006 S008 · impliesBefore arbitrary x in R^3;all column sums<=12 After norm1(Ax)<=sum_j(sum_i abs(Aij))*abs(xj)<=12*norm1(x) Conditions: none Apply the triangle inequality to each output,exchange finite sums,and bound every column coefficient by L.This yields one constant for every real input,not samples. Step identifiers and dependenciesRule: operator_l1_universal_column_bound · Uses: G001, S007 S009 · impliesBefore attaining unit input image:matrix=[[9, -3, 3], [-1, -9, -5]];input=[0, 1, 0] After input dimension=3;output dimension=2;rows dot input Conditions: none Retain column-input/row-output roles for rectangular matrices.Computed values can exceed raw input bounds and are not clipped. Step identifiers and dependenciesRule: matrix_row_column_roles · Uses: G001, S008 S010 · calculationBefore attaining unit input image row0 column0 product := mul(9, 0) After attaining unit input image row0 column0 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore attaining unit input image row0 column1 product := mul(-3, 1) After attaining unit input image row0 column1 product = -3 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore attaining unit input image row0 column2 product := mul(3, 0) After attaining unit input image row0 column2 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore attaining unit input image row0 sum := sum(0, -3, 0) After attaining unit input image row0 sum = -3 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore attaining unit input image row1 column0 product := mul(-1, 0) After attaining unit input image row1 column0 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore attaining unit input image row1 column1 product := mul(-9, 1) After attaining unit input image row1 column1 product = -9 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore attaining unit input image row1 column2 product := mul(-5, 0) After attaining unit input image row1 column2 product = 0 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore attaining unit input image row1 sum := sum(0, -9, 0) After attaining unit input image row1 sum = -9 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · impliesBefore attaining input:signed vector=[0, 1, 0];norm=l1 After coordinate absolute values=[0, 1, 0] Conditions: none Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds. Step identifiers and dependenciesRule: operator_named_norm_original_signs · Uses: G001, S017 S019 · calculationBefore attaining input l1 sum := sum(0, 1, 0) After attaining input l1 sum = 1 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore attaining output:signed vector=[-3, -9];norm=l1 After coordinate absolute values=[3, 9] Conditions: none Take absolute values of whole coordinates without changing their pairing.This also accepts computed matrix images without clipping to raw input bounds. Step identifiers and dependenciesRule: operator_named_norm_original_signs · Uses: G001, S019 S021 · calculationBefore attaining output l1 sum := sum(3, 9) After attaining output l1 sum = 12 Conditions: none Compute exactly from original roles and established intermediates,retaining coordinates,coefficients,signs and necessary guards. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · impliesBefore explicit input=[0, 1, 0];input norm=1;image=[-3, -9];output norm=12 After uniform upper bound=12 and attained ratio=12;exact induced norm=12 Conditions: none For l1 take the maximizing column's basis vector;for linfinity take signs of a maximizing row.For a zero matrix still choose a nonzero unit vector.The universal bound L and an attained unit ratio L exclude every smaller constant,establishing the exact norm. Step identifiers and dependenciesRule: operator_attainment_proves_least_uniform_constant · Uses: G001, S021 ## 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 ([[9, -3, 3], [-1, -9, -5]], [10, 12, 8], 12, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 71adc41b11527e16a5938118b4d009e5688005cedd15d0d4b495a7a07fb1ccab - candidate record SHA-256: 169eead665fa58e66a72056ef3b4d45beb507a353250e520fa42730f1c748230 - semantic review receipt SHA-256: f335caca08574c9f949c141083c66655fee51c227d45d80a3ddb7d78f862d798 - question-first witness SHA-256: cc7ba54738af91a6871fc1c20e45ad8f2b434392273fa334bd16a3b2238f4a77 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261028-r05-bounded-linear-operator-0022.json)