Computation of Eigenvalues and Eigenvectors Using C (Linear Algebra: Linear Transformation and Diagonalization)

Linear Algebra

PROGRAMMES IN PYTHON AND C LANGUAGES

UNIT − II: LINEAR TRANSFORMATION AND DIAGONALIZATION

COMPUTATION OF EIGENVALUES AND EIGENVECTORS USING C

The computation of eigenvalues and eigenvectors for a 3×3 matrix in C involves several mathematical steps and numerical methods.

1. Finding eigenvalues

• Characteristic equation: For a matrix A, the eigenvalues (λ) are found by solving the characteristic equation: det (A−λI) = 0, where I is the identity matrix. For a 3×3 matrix, this results in a cubic polynomial in λ.

• Solving the cubic equation: Solving a cubic equation can be done numerically using methods like Newton−Raphson or analytically using Cardano's method. Since analytical solutions can be complex, numerical methods are often preferred for their practical implementation in C.

2. Find eigenvectors

• For each eigenvalue: Once an eigenvalue λ is found, substitute it back into the equation (A−λI)x=0, where x is the eigenvector.

• Solving the system of equations: This forms a system of linear equations. Since the determinant of (A − λI) is zero, this system will have non−trivial solutions (i.e., non−zero eigenvectors).

• Gaussian Elimination or Row Echelon form: Use techniques like Gaussian elimination or converting the matrix to row echelon form to find the null space of (A−λI), which represents the eigenvectors corresponding to that eigenvalue. The resulting solutions will be in terms of free variables, defining the direction of the eigenvector.

C Implementation considerations

• Matrix representation: Represent the 3×3 matrix using a 2D array (e.g., float matrix [3][3];).

• Determinant calculation: Implement a function to calculate the determinant of a 3×3 matrix.

• Gaussian Elimination/Row Reduction: Implement functions to perform row operations for Gaussian elimination to find the eigenvectors.

• Floating−point precision: Be mindful of floating−point precision issues when comparing values to zero or performing calculations involving small numbers.

Example structure (Conceptual C code)

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#include <stdio.h>

#include <math.h>

// Function to calculate determinant of a 3×3 matrix

float determinant_3 x 3 (float matrix[3] [3]

{

// Implement determinant calculation

}

// Function to solve a cubic equation numerically (e.g., Newton−Raphson)

// Returns the roots (eigenvalues)

void solve_cubic_equation (float a, float b, float c, float d, float * roots)

{

// Implement numerical method to find roots

}

// Function to perform Gaussian elimination and find eigenvectors

void find_eigenvector (float matrix [3] [3], float eigenvalue, float* eigenvector)

{

// Construct (A - Lambda I)

// Perform Gaussian elimination

// Solve for eigenvector components

}

int main() {

float A[3] [3] = {

{a11, a12, a13),

(a21, a22, a23},

(a31, a32, a33)

}

// 1. Form the characteristic polynomial coefficients

/ Based on det (A ‒ Lambda I) = ‒ Lambda^3 + S1*Lambda^2 − S2*Lambda + S3 =

// Where S1 = trace (A), S2 = sum of principal minors, S3 = det (A)

// 2. Solve the cubic equation for eigenvalues float eigenvalues [3];

for (int 1−0; 1 < 3; 1++)

{

float eigenvector [3];

find_eigenvector (A, eigenvalues [i], eigenvector);

//print or store the eigenvector

}

return 0;

}