2×2 Matrix Characteristic Polynomial, Eigenvalues (λ₁, λ₂), Trace & Condition Number Calculator

Solve λ² − (a+d)λ + (ad−bc) = 0 for eigenvalues λ₁, λ₂, trace, and determinant.

Dominant Eigenvalue (λ₁)
6 λ₁
Second Eigenvalue (λ₂) 1 λ₂
Matrix Trace Tr(A) = λ₁ + λ₂ 7 trace
Matrix Determinant det(A) = λ₁·λ₂ 6 det
Inputs
⚡ Scenarios:
a₁₁
Top-left entry a₁₁.
a₁₂
Top-right entry a₁₂.
a₂₁
Bottom-left entry a₂₁.
a₂₂
Bottom-right entry a₂₂.
Results & Graph
Dominant Eigenvalue (λ₁)
6 λ₁
Second Eigenvalue (λ₂)
1 λ₂
Matrix Trace Tr(A) = λ₁ + λ₂
7 trace
Matrix Determinant det(A) = λ₁·λ₂
6 det

Dominant Eigenvalue (λ₁)

Graph 1 • 2×2 Matrix Eigenvalues (λ₁, λ₂) • Domain Architecture & Live Telemetry Maps Matrix A11, Matrix A12, Matrix A21 through 2×2 Matrix Eigenvalues (λ₁, λ₂) Formula.
Dominant Eigenvalue (λ₁): 6 λ₁ 2×2 Matrix Eigenvalues (λ₁, λ₂) • Invariants & Metrics Dominant Eigenvalue (λ₁) 6 λ₁ Second Eigenvalue (λ₂) 1 λ₂ Matrix Trace Tr(A) = λ₁ + λ₂ 7 trace Matrix Determinant det(A) = 6 det
Graph 2 • Dominant Eigenvalue (λ₁) Across Matrix A11 Sweep (40% → 160%) Evaluates Tr(A) = a + d; det(A) = a·d − b·c; λ₁,₂ = [ Tr(A) ± √(Tr(A)² − 4·det across 7 operating points (current input highlighted).
0 λ₁ 4.73 λ₁ 9.47 λ₁ 4 2a₁₁ 4.56 3a₁₁ 5.24 4a₁₁ 6 5a₁₁ 6.83 6a₁₁ 7.70 7a₁₁ 8.61 8a₁₁
Graph 3 • Joint Sensitivity: Matrix A11 × Matrix A12 Compares Dominant Eigenvalue (λ₁) when Matrix A12 is shifted +25% (solid) vs. −25% (dashed).
0 λ₁ 4.08 λ₁ 8.16 λ₁ --- Matrix A12 @ 75% 5.11 -30% 5.63 -15% 6.19 +0% 6.79 +15% 7.42 +30%
Summary: For A = [[5, 2], [2, 2]], Trace = 7, Determinant = 6, and eigenvalues are λ₁ = 6 and λ₂ = 1 (spectral condition ratio 6).
Summary: For A = [[5, 2], [2, 2]], Trace = 7, Determinant = 6, and eigenvalues are λ₁ = 6 and λ₂ = 1 (spectral condition ratio 6).
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Formula & Mathematical Methodology: 2×2 Matrix Eigenvalues (λ₁, λ₂) Formula

Tr(A) = a + d
det(A) = a·d − b·c
λ₁,₂ = [ Tr(A) ± Tr(A² − 4·det(A)) ] / 2
Tr(A) = a + d; det(A) = a·d − b·c; λ₁,₂ = [ Tr(A) ± √(Tr(A)² − 4·det(A)) ] / 2
  • matrixA11: Matrix A11 (a₁₁)
  • matrixA12: Matrix A12 (a₁₂)
  • matrixA21: Matrix A21 (a₂₁)
  • matrixA22: Matrix A22 (a₂₂)

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Worked Mathematical Example: 2×2 Matrix Eigenvalues (λ₁, λ₂)

Solve λ² − (a+d)λ + (ad−bc) = 0 for eigenvalues λ₁, λ₂, trace, and determinant.

  1. Specify the function parameters, matrix entries, or confusion matrix counts.
  2. Apply the closed-form or iterative formula: Tr(A) = a + d; det(A) = a·d − b·c; λ₁,₂ = [ Tr(A) ± √(Tr(A)² − 4·det(A)) ] / 2.
  3. Examine the primary result, error bounds, and 3 interactive SVG plots.
Result: Computes 2×2 Matrix Eigenvalues (λ₁, λ₂) via Tr(A) = a + d; det(A) = a·d − b·c; λ₁,₂ = [ Tr(A) ± √(Tr(A)² − 4·det(A)) ] / 2.

Frequently Asked Questions

How does the 2×2 Matrix Characteristic Polynomial, Eigenvalues (λ₁, λ₂), Trace & Condition Number Calculator compute the solution?

It evaluates Tr(A) = a + d; det(A) = a·d − b·c; λ₁,₂ = [ Tr(A) ± √(Tr(A)² − 4·det(A)) ] / 2 directly in your browser and updates three custom SVG visualizations in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this calculator run locally offline?

Yes. All calculations run 100% locally in your browser and work offline via the CalculatorGraph PWA.

Mathematics & Statistics · Linear Algebra & Vectors

2×2 Matrix Characteristic Polynomial, Eigenvalues (λ₁, λ₂), Trace & Condition Number Calculator: inputs, method, and how to read the result

Solve λ² − (a+d)λ + (ad−bc) = 0 for eigenvalues λ₁, λ₂, trace, and determinant.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Matrix A11: Top-left entry a₁₁.
  • Matrix A12: Top-right entry a₁₂.
  • Matrix A21: Bottom-left entry a₂₁.
  • Matrix A22: Bottom-right entry a₂₂.

What the calculation includes

With the default values shown, Dominant Eigenvalue (λ₁) is 6 λ₁. The calculation follows 2×2 Matrix Eigenvalues (λ₁, λ₂) Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Matrix A11 from 5 a₁₁ to 6 a₁₁ changes Dominant Eigenvalue (λ₁) from 6 λ₁ to 6.828 λ₁. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses exact analytical or standard numerical analysis formulations from NIST DLMF.
  • Double-precision floating-point arithmetic has ~15–17 significant decimal digits of precision.

Scope: Global Mathematical & Statistical Standard Check the assumptions and limitations against your situation before relying on the estimate.

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