Computation of Null Space and Range of Matrix by Using C (Linear Algebra: Linear Transformation and Diagonalization)

Linear Algebra

PROGRAMMES IN PYTHON AND C LANGUAGES

UNIT − II: LINEAR TRANSFORMATION AND DIAGONALIZATION

COMPUTATION OF NULL SPACE AND RANGE OF MATRIX BY USING C

The computation of the null space and range of a matrix in C involves implementing Gaussian elimination to achieve Reduced Row Echelon Form (RREF) and then interpreting the results.

1. Null Space (Kernel)

• Represent the Homogeneous System: Given a matrix A, the null space consists of all vectors x such that Ax=0. This is represented by an augmented matrix [A|0], where 0 is a column vector of zeros.

• Gaussian Elimination to RREF: Implement a C function to perform Gaussian elimination on the augmented matrix to bring it to its RREF.

The involves:

• Finding pivots (leading non−zero entries)

• Making pivots equal to 1.

• Eliminating entries above and below pivots.

• Swapping rows if necessary to bring pivot rows to the top.

• Identify free variables: In the RREF, columns without a leading 1 (pivot) correspond to free variables.

• Express pivot variables in terms of free variables: Write the system of equations from the RREF. Solve for the pivot variables in terms of the free variables.

• Construct basis vectors: For each free variable, set it to 1 and all other free variables to 0. This generates a basis vector for the null space. The collection of these vectors forms a basis for the null space.

2. Range (column space)

• Identify pivot columns: After obtaining the RREF of the original matrix A, the columns in the original matrix A that correspond to the pivot columns in the RREF form a basis for the range (column space).

• Construct basis: Collect these corresponding columns from the original matrix A to form a basis for the range.

C Implementation Considerations

• Matrix representation: Use a 2D array (e.g., double** matrix) to represent the matrix.

• Dynamic memory allocation: Allocate memory dynamically for the matrix, especially for varying sizes.

• Floating−point precision: Be mindful of floating−point precision issues when comparing values to zero during Gaussian elimination. Use a small epsilon value for comparisons.

• Function design: Create separate functions for

• Matrix initialization and input.

• Gaussian elimination to RREF.

• Extracting null space basis vectors.

• Extracting range basis vectors.

• Matrix printing for debugging.

Example (Conceptual C Snippet for Gaussian Elimination)

C

// Simplified example, actual implementation requires robust error handling and prevoid

gaussianElimination (double** matrix, int rows int cols)

{

int pivotRow = 0;

for (int j =0; j< cols && pivot Row < rows; ++j)

{

// Find pivot

in tmaxRow = pivotRow;

for (int i = pivot Row +1; i < rows; ++i)

{

  if(fabs (matrix[i][j]) > fabs (matrix [maxRow] [j]))

{

   maxRow=i;

}

}

// Swap rows if necessary

if (maxRow != pivotRow)

{

 // Swap matrix [pivotRow] and matrix [maxRow]

}

// Make pivot 1 and eliminate other entries in the column

if (fabs (matrix [pivotRow] [j]) > 1e−9)

{

//Check non−zero pivot.

double pivotVal = matrix [pivotRow][j];

for (int i=0; i < rows; ++i)

{

if (i != pivotRow) {

double factor = matrix[i][j]];

for (int k = j; k < cols; ++k) {

matrix[i] [k] ‒= factor * matrix [pivotRow] [k];

}

}

}

pivotRow++;

}

}

}