Math calculator

Condition Number Calculator

κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix. Same engine as Linear System. Singular matrices return infinite. Not a 2-norm product. Runs locally.

Instant result

Result

—

Enter values to calculate.

Inputs—

Mode—

Formula—

Trust summary CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes

Input interpretation: Enter values to calculate.
Result: —
Verified scope: all published modes
Assurance: Core
Declared partition coverage: PASS · 19/19 declared partitions (solve, det, inverse, multiply, transpose, rank, eigen, lu, chol, qr, svd, lstsq, rref, condition, nullspace, columnspace, rowspace, leftnullspace, invalid-domain) · Matrix
Known limitations: Core CVP does not include live graph, viewport, or pointer interaction.
Model: κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix (order 2–4).
Scope: Real square matrices
Verification: Engine tested · Source checked · v1.14.16 · CVP VERIFIED · CVP protocol 1.0.0-proposed · Core assurance · all published modes· View Manifest · CVP overview · Specification
Versions: Calculation 1.14.16 · CVP protocol 1.0.0-proposed · Evidence 2026-10-30.cholesky-numeric-contract-v2
Verification revision: 2026-10-30.cholesky-numeric-contract-v2 · 10/10 property · digest 2f8af10fde34
Legacy regression: 58/58 tests · Production surface contract 6/6
Trust layers: Verification VERIFIED · Production CURRENT · Overall CVP state: VERIFIED under CVP 1.0.0-proposed
Reference: O1 model · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 expected_values · O2 numerical_behavior
Interfaces: PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 24/24 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Raw HTTP REST/JSON-RPC transport rejection 5/5; raw-number source and conflicting pins are transport-only checks, not engine/SSR input-source parity.
Supplemental domain review: Not performed
Named expert review: Not performed
CVP suite: 478/478 golden · 8/8 CVP boundary · 78/78 invalid · 10/10 property · 2/2 metamorphic · 3/3 round-trip · 67/67 cross-interface · 2/2 cross-calculator · 6/6 CVP contract · Manifest
Sources: ISO 80000-2:2019 Quantities and units — Mathematics
NIST DLMF §3.2 — Linear Algebra — §3.2
Sources
Evidence: 29 legacy golden · 6 legacy boundary · legacy regression suite · 478/478 oracle-backed golden · 78/78 invalid · Artifact integrity PASS
Semantic contract: PASS

Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

1-norm conditionκ₁(A) = ‖A‖₁ ‖A⁻¹‖₁

Singulardet(A) = 0 ⇒ κ₁ = ∞

iSquare matrices only, order ≤ 4. Reuses the 1-norm already used as a trust field. Ill-conditioned warning when κ₁ ≥ 10⁸. Not a CAS. Not 2-norm / Frobenius κ. Float64.

How to use

Enter a square matrix

2×2 or 3×3. Larger composition lives at /workspace/matrix. Inverse and rank are sibling discovery URLs.

Read κ₁

Finite when A is invertible. Singular matrices return infinite, not SINGULAR_MATRIX.

Example calculations

Common configurations with formula and result.

Diagonal

diag(2, 1)

‖A‖₁=2, ‖A⁻¹‖₁=1

κ₁=2

Singular

[[1,2],[2,4]]

det=0

∞

Condition Number calculator specification

Version 1.14.16 · Engine tested

Calculation status

Engine tested 58/58 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.14.16

Review policy · Evidence

Definition: κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁. This page is mode=condition on math.linear_algebra. The same κ₁ already appears as a trust field on other modes. Not a second engine. Not κ₂ = σ_max/σ_min as a product.
What it calculates: κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁ for a small real square matrix (order 2–4).
Inputs: order
a11…
Outputs: condition_number
infinite
status
det
Formula: κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁
Assumptions: Real square matrices
Order ≤ 4
Planar ℝ² figure when order=2; order 3+ omits it
Float64
Not a CAS
Not 2-norm κ₂ / Jordan form
Units: dimensionless
Boundary conditions: singular A → condition_number null, infinite true
unknown mode (including jordan-form) → INVALID_MODE
Example: diag(2,1) → κ₁=2
Validation cases: 2 published on this page · 58/58 tests · Production surface contract 6/6 · View evidence
order=2 a11=2 a12=0 a21=0 a22=1 → condition_number=2
mode=jordan-form order=2 a11=1 a12=0 a21=0 a22=1 → INVALID_MODE
Sources: ISO 80000-2:2019 Quantities and units — Mathematics
Supports: Mathematical symbols and notation. This standard is not numerical verification or an endorsement of this implementation.
NIST DLMF §3.2 — Linear Algebra — §3.2
Supports: Gaussian elimination, LU decomposition, conditioning, eigenvalues, and eigenvectors.
Calculation version: 1.14.16

Background

Interpretation and common distinctions.

What does this condition number calculator compute?

κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁.

The published formula or method is κ₁(A) = ‖A‖₁ ‖A⁻¹‖₁. The following references support the stated mathematics or method:

NIST DLMF §3.2 — Linear Algebra: Gaussian elimination, LU decomposition, conditioning, eigenvalues, and eigenvectors.

Which inputs and limits should you check?

Rank, inverse, subspace, and least-squares decisions depend on the stated tolerance and conditioning. Inspect residuals and warnings before interpreting a nearly singular matrix.

The worked example uses diag(2, 1) with ‖A‖₁=2, ‖A⁻¹‖₁=1. Compare it with the Example calculations and the complete result for the same inputs.

How can you verify and cite this result?

CalculatorX is responsible for this page. The mathematical references above explain the model; implementation checks and their scope are documented separately in the math.linear_algebra v1.14.16 Evidence and capability contract. Read the displayed verification and production state together with the specification's numerical limits.

Use the clean calculator URL to cite the tool. A shared input link identifies a particular calculation. Content updates, numerical verification, and domain reviews have separate dates and scopes.

Input replay and numerical range

All entries share math.linear_algebra v1.14.16. Enter every cell as a finite decimal, including scientific notation. Empty cells and nonzero decimals below the binary64 range are rejected. QR, SVD, and least squares accept dimensions from 2 to 4; other operations require square matrices. Matching aliases may be supplied, while conflicting aliases or nested and flat cells are rejected.

A singular system may be consistent with infinitely many solutions or inconsistent; read the status and warnings. Tolerance-based rank and subspace decisions are not exact symbolic certificates. Results outside the numerical range refuse with NUMERIC_PRECISION_LIMIT. An unavailable auxiliary determinant is omitted with a warning, rather than reported as zero or null.

Calculations run locally in the browser. Shared query links contain inputs, are sent to the server when opened, and may appear in server or third-party logs. Anyone receiving a shared link can read its inputs. JSON and CSV exports include the original inputs, complete structured result, warnings, and calculation version.

Other calculators in this family: Cholesky Factorization Calculator, Column Space Calculator, Least Squares Calculator, Left Nullspace Calculator, Linear System Calculator, LU Factorization Calculator, Matrix Determinant Calculator, Matrix Eigenvalues Calculator . Explore all Linear Algebra.

Frequently asked questions

Key distinctions behind the calculation.

Is this a second linear-algebra engine?

No. It is mode=condition on math.linear_algebra, the same engine as /calc/math/linear-system and /workspace/matrix. κ₁ already exists as a trust field on other modes.

Is this the 2-norm condition number σ_max/σ_min?

No. This page reports the 1-norm κ₁. SVD lives at /calc/math/matrix-svd. A κ₂ product stays later on this same engine.