Eigenvector Calculator

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Eigenvector Calculator (2×2)
2×2 Square Matrix
Enter Matrix Elements [A]:
Results:
function calculateResult(){var a11=parseFloat(document.getElementById('a11').value);var a12=parseFloat(document.getElementById('a12').value);var a21=parseFloat(document.getElementById('a21').value);var a22=parseFloat(document.getElementById('a22').value);var showNorm=document.getElementById('steps').checked;if(isNaN(a11)||isNaN(a12)||isNaN(a21)||isNaN(a22)){alert('Please fill in all matrix fields with valid numbers.');return;}var trace=a11+a22;var det=(a11*a22)-(a12*a21);var discriminant=Math.pow(trace,2)-(4*det);if(discriminant<0){document.getElementById('eigenValuesResult').innerHTML="Eigenvalues: Complex roots detected (Real-world eigenvector calculators primarily handle real solutions for simple matrices).";document.getElementById('eigenVectorsResult').innerHTML="";return;}var l1=(trace+Math.sqrt(discriminant))/2;var l2=(trace-Math.sqrt(discriminant))/2;function getVector(l,a11,a12,a21,a22){var x,y;if(a12!==0){x=a12;y=l-a11;}else if(a21!==0){x=l-a22;y=a21;}else{x=1;y=0;}if(showNorm){var mag=Math.sqrt(x*x+y*y);x=x/mag;y=y/mag;}return [x.toFixed(4),y.toFixed(4)];}var v1=getVector(l1,a11,a12,a21,a22);var v2=getVector(l2,a11,a12,a21,a22);var valStr="Eigenvalues:
λ1 = "+l1.toFixed(4)+"
λ2 = "+l2.toFixed(4);var vecStr="Corresponding Eigenvectors:
v1 = ["+v1[0]+", "+v1[1]+"]T
v2 = ["+v2[0]+", "+v2[1]+"]T";document.getElementById('eigenValuesResult').innerHTML=valStr;document.getElementById('eigenVectorsResult').innerHTML=vecStr;}

Calculator Use

The Eigenvector Calculator is a specialized tool designed to find the characteristic values (eigenvalues) and their associated vectors (eigenvectors) for a 2×2 square matrix. In linear algebra, eigenvectors are non-zero vectors that only change by a scalar factor when a linear transformation is applied to them.

This tool is essential for students and engineers working with systems of linear equations, vibration analysis, or principal component analysis (PCA). Simply input the four coefficients of your 2×2 matrix to receive the results instantly.

Matrix Elements (a11, a12, a21, a22)
These represent the individual numbers within your square matrix [A].
Eigenvalues (λ)
The scalar values that represent how much the eigenvector is scaled during the transformation.
Eigenvectors (v)
The vectors that maintain their direction after the matrix multiplication.

How It Works

To find the eigenvectors of a matrix, you must follow a two-step mathematical process. First, find the eigenvalues, and then solve for the vectors. The fundamental equation is:

A v = λ v

Where A is the matrix, v is the eigenvector, and λ (lambda) is the eigenvalue. The calculator performs the following:

  • Step 1: The Characteristic Equation. We solve det(A – λI) = 0. For a 2×2 matrix, this results in a quadratic equation: λ² – Tr(A)λ + det(A) = 0.
  • Step 2: Solve for λ. Using the quadratic formula, we find the roots λ1 and λ2.
  • Step 3: Solve the Null Space. For each eigenvalue, we solve the system (A – λI)v = 0 to find the components of vector v.
  • Step 4: Normalization (Optional). If selected, the calculator scales the vector so its total length equals 1.

Calculation Example

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

Step-by-step solution:

  1. Calculate Trace: 4 + 3 = 7.
  2. Calculate Determinant: (4 * 3) – (1 * 2) = 10.
  3. Characteristic Equation: λ² – 7λ + 10 = 0.
  4. Factor Equation: (λ – 5)(λ – 2) = 0. Thus, λ1 = 5, λ2 = 2.
  5. For λ1 = 5: Solve (4-5)x + 1y = 0 → -x + y = 0. One solution is v1 = [1, 1].
  6. For λ2 = 2: Solve (4-2)x + 1y = 0 → 2x + y = 0. One solution is v2 = [1, -2].

Common Questions

Can a matrix have no eigenvectors?

Every n x n matrix has at least one eigenvalue (though they may be complex). If the eigenvalues are complex numbers, the eigenvectors will also have complex components.

What is a normalized eigenvector?

Since any scalar multiple of an eigenvector is also an eigenvector, mathematicians often "normalize" them. This means scaling the vector so its magnitude (length) is exactly 1, making it a "unit vector."

Why are eigenvectors useful in real life?

They are used in Google's PageRank algorithm to rank websites, in bridge design to find resonance frequencies (to prevent collapse), and in data science to reduce the dimensions of large datasets via PCA.

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