Eigenvalue Calculator for 2×2 Matrices
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The online Eigenvalue Calculator for 2×2 Matrices helps you calculate instantly and solve problems related to Eigenvalues Eigenvectors. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Find the eigenvalues of a 2×2 matrix from the characteristic polynomial p(λ)=det(λI−A). Shows trace, determinant, discriminant, and real or complex conjugate roots.
p(λ)=λ² − tr(A)λ + det(A); λ = [tr ± √(tr² − 4 det)] / 2
For a 2×2 matrix A=[[a,b],[c,d]], the eigenvalues are the roots of the monic characteristic polynomial p(λ)=det(λI−A). The discriminant Δ=tr²−4det classifies distinct real, repeated, or complex conjugate eigenvalues.
- Matrix A is exactly 2×2 with real entries.
- Convention is p(λ)=det(λI−A).
- Complex eigenvalues are supported and shown as a conjugate pair when Δ<0.
Inputs
Results
Worked Examples
Distinct real eigenvalues
[[4,2],[1,3]] → λ=5 and λ=2 (p(λ)=λ²−7λ+10).
- matrix: [ [ 4, 2 ], [ 1, 3 ] ]
- eigenvalues: λ₁ = 5, λ₂ = 2
- trace: 7
- determinant: 10
Repeated eigenvalue
[[1,1],[0,1]] → λ=1 with algebraic multiplicity 2.
- matrix: [ [ 1, 1 ], [ 0, 1 ] ]
- eigenvalues: λ = 1 (repeated; algebraic multiplicity 2)
- multiplicity: 2
Complex conjugate pair
[[0,-1],[1,0]] → λ=i and λ=−i (p(λ)=λ²+1).
- matrix: [ [ 0, -1 ], [ 1, 0 ] ]
- eigenvalues: λ₁ = i, λ₂ = −i (complex conjugate pair)
- characteristic_polynomial: λ² + 1
About this calculator
Overview
The Eigenvalue Calculator for 2×2 Matrices finds the eigenvalues of a real 2×2 matrix A by solving the characteristic equation det(λI − A)=0. It reports the eigenvalues, trace, determinant, discriminant, and the monic characteristic polynomial, including repeated roots and complex conjugate pairs.
When to use
Use this tool when you need the spectrum of a 2×2 matrix for stability checks, modal analysis, diagonalization prep, or verifying hand calculations of λ from tr(A) and det(A).
Inputs explained
- Matrix A — exactly 2×2 real matrix (four editable cells)
Formula / method
Compute tr(A)=a+d and det(A)=ad−bc. Form p(λ)=λ² − tr(A)λ + det(A). The discriminant Δ=tr² − 4 det classifies the roots: Δ>0 distinct real, Δ=0 repeated real (multiplicity 2), Δ<0 complex conjugates. Roots are λ=[tr ± √Δ]/2 (or ±i√(−Δ) when complex).
Worked example
For A=[[4,2],[1,3]], tr=7, det=10, p(λ)=λ²−7λ+10, Δ=9 → eigenvalues 5 and 2.
Interpreting results
Eigenvalues are the scalars λ such that Av=λv for some nonzero eigenvector v. Their sum equals the trace and their product equals the determinant. Complex eigenvalues of a real matrix always appear as conjugate pairs.
Assumptions
- Entries are real numbers in a 2×2 grid.
- Characteristic polynomial uses the monic convention det(λI − A).
- Floating-point rounding may affect near-zero discriminants.
Limitations
- This page is limited to 2×2 matrices; larger sizes need other eigenvalue tools.
- Eigenvectors are computed on the companion Eigenvector 2×2 Calculator.
- Educational numerical tool — verify critical designs independently.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Compute tr(A)=a+d and det(A)=ad−bc. Form p(λ)=λ² − tr(A)λ + det(A). The discriminant Δ=tr² − 4 det classifies the roots: Δ>0 distinct real, Δ=0 repeated real (multiplicity 2), Δ<0 complex conjugates. Roots are λ=[tr ± √Δ]/2 (or ±i√(−Δ) when complex).
Example Calculation
For A=[[4,2],[1,3]], tr=7, det=10, p(λ)=λ²−7λ+10, Δ=9 → eigenvalues 5 and 2.
Frequently Asked Questions
What is Eigenvalue Calculator for 2×2 Matrices?
- The Eigenvalue Calculator for 2×2 Matrices finds the eigenvalues of a real 2×2 matrix A by solving the characteristic equation det(λI − A)=0. It reports the eigenvalues, trace, determinant, discriminant, and the monic characteristic polynomial, including repeated roots and complex conjugate pairs.
How does Eigenvalue Calculator for 2×2 Matrices work?
- Compute tr(A)=a+d and det(A)=ad−bc. Form p(λ)=λ² − tr(A)λ + det(A). The discriminant Δ=tr² − 4 det classifies the roots: Δ>0 distinct real, Δ=0 repeated real (multiplicity 2), Δ<0 complex conjugates. Roots are λ=[tr ± √Δ]/2 (or ±i√(−Δ) when complex).
Why use this Engineering Math calculator?
- Use this tool when you need the spectrum of a 2×2 matrix for stability checks, modal analysis, diagonalization prep, or verifying hand calculations of λ from tr(A) and det(A).