Eigenvalue Calculator

by
Eigenvalue Calculator
2×2 Matrix
Enter Matrix Elements:
Results:
Enter values and click Calculate to see the eigenvalues (λ).
function calculateEigenvalues(){var a11=parseFloat(document.getElementById('a11').value)||0;var a12=parseFloat(document.getElementById('a12').value)||0;var a21=parseFloat(document.getElementById('a21').value)||0;var a22=parseFloat(document.getElementById('a22').value)||0;var showSteps=document.getElementById('showSteps').checked;var trace=a11+a22;var determinant=(a11*a22)-(a12*a21);var discriminant=(trace*trace)-(4*determinant);var answerDiv=document.getElementById('answer');var resultHtml=";if(discriminant>0){var lambda1=(trace+Math.sqrt(discriminant))/2;var lambda2=(trace-Math.sqrt(discriminant))/2;resultHtml='
λ₁ = '+lambda1.toFixed(4)+'
λ₂ = '+lambda2.toFixed(4)+'
';}else if(discriminant===0){var lambda=trace/2;resultHtml='
λ₁ = λ₂ = '+lambda.toFixed(4)+' (Repeated Eigenvalue)
';}else{var realPart=(trace/2).toFixed(4);var imagPart=(Math.sqrt(-discriminant)/2).toFixed(4);resultHtml='
λ₁ = '+realPart+' + '+imagPart+'i
λ₂ = '+realPart+' – '+imagPart+'i
';}if(showSteps){resultHtml+='
';resultHtml+='Step-by-Step Solution:
';resultHtml+='1. Characteristic Equation: det(A – λI) = 0
';resultHtml+='2. |'+a11+' – λ   '+a12+'|
    |'+a21+'   '+a22+' – λ| = 0
';resultHtml+='3. ('+a11+' – λ)('+a22+' – λ) – ('+a12+' * '+a21+') = 0
';resultHtml+='4. λ² – ('+trace+')λ + ('+determinant+') = 0
';resultHtml+='5. Using Quadratic Formula: λ = [-b ± sqrt(b² – 4ac)] / 2a
';resultHtml+='6. λ = ['+trace+' ± sqrt(('+trace+')² – 4(1)('+determinant+'))] / 2
';resultHtml+='7. λ = ['+trace+' ± sqrt('+discriminant.toFixed(2)+')] / 2
';}answerDiv.innerHTML=resultHtml;}

Calculator Use

This eigenvalue calculator helps you find the characteristic roots of a 2×2 square matrix. Eigenvalues are a fundamental concept in linear algebra, representing the factors by which a vector is scaled during a linear transformation. This tool is essential for students and engineers working with differential equations, structural analysis, or machine learning algorithms like Principal Component Analysis (PCA).

To use the calculator, simply enter the four components of your 2×2 matrix into the corresponding fields (a₁₁, a₁₂, a₂₁, and a₂₂). The calculator will handle real, repeated, and complex eigenvalues automatically.

Matrix Elements (a₁₁, a₁₂, etc.)
These are the numerical values of the matrix. For a 2×2 matrix, the first row contains a₁₁ and a₁₂, and the second row contains a₂₁ and a₂₂.
Show Solution Steps
Enable this checkbox to see the mathematical breakdown of the characteristic equation and the quadratic formula application.

How It Works

When you find the eigenvalues of a matrix A, you are solving for the values of λ that satisfy the characteristic equation:

det(A – λI) = 0

For a 2×2 matrix, this expands into a quadratic equation:

  • Trace: The sum of the diagonal elements (a₁₁ + a₂₂).
  • Determinant: The product of the diagonals minus the product of the anti-diagonals (a₁₁a₂₂ – a₁₂a₂₁).
  • Characteristic Polynomial: λ² – (Trace)λ + (Determinant) = 0.

The roots of this polynomial are the eigenvalues. If the discriminant (Trace² – 4 * Determinant) is negative, the eigenvalues will be complex numbers involving 'i'.

Calculation Example

Example: Find the eigenvalues for the matrix A = [[4, 1], [2, 3]].

Step-by-step solution:

  1. Identify elements: a₁₁=4, a₁₂=1, a₂₁=2, a₂₂=3
  2. Calculate Trace: 4 + 3 = 7
  3. Calculate Determinant: (4 * 3) – (1 * 2) = 12 – 2 = 10
  4. Form Polynomial: λ² – 7λ + 10 = 0
  5. Factor the quadratic: (λ – 5)(λ – 2) = 0
  6. Result: λ₁ = 5, λ₂ = 2

Common Questions

What are eigenvalues used for?

Eigenvalues are used to simplify complex linear transformations. They are crucial in stability analysis (determining if a system will remain stable over time), vibration analysis in mechanical engineering, and reducing data dimensionality in statistics.

Can a matrix have zero as an eigenvalue?

Yes. If zero is an eigenvalue, it means the determinant of the matrix is zero, which implies the matrix is singular (not invertible).

What if the eigenvalues are complex?

Complex eigenvalues indicate a rotation in the transformation. In the context of dynamical systems, they often represent oscillatory behavior or spirals in a phase portrait.

Leave a Comment