A square matrix as sum of symmetric and skew-symmetric matrix ?

A square matrix can be expressed as the sum of a symmetric matrix and a skew-symmetric matrix. This decomposition is unique and provides insight into the structure of any square matrix.

Symmetric Matrix − A matrix whose transpose is equal to the matrix itself (A = A^T).

Skew-symmetric Matrix − A matrix whose transpose is equal to the negative of the matrix (A = -A^T).

Syntax

A = (1/2)(A + A^T) + (1/2)(A - A^T)

Where:

• (1/2)(A + A^T) is the symmetric part
• (1/2)(A - A^T) is the skew-symmetric part

Example

This program decomposes a 3x3 matrix into its symmetric and skew-symmetric components −

#include <stdio.h>

#define N 3

void printMatrix(float mat[N][N]) {
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++)
            printf("%.1f ", mat[i][j]);
        printf("<br>");
    }
}

int main() {
    float mat[N][N] = { { 2, -2, -4 },
                        { -1, 3, 4 },
                        { 1, -2, -3 } };
    
    float tr[N][N];
    
    /* Calculate transpose */
    for (int i = 0; i < N; i++)
        for (int j = 0; j < N; j++)
            tr[i][j] = mat[j][i];
    
    float symm[N][N], skewsymm[N][N];
    
    /* Calculate symmetric and skew-symmetric parts */
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < N; j++) {
            symm[i][j] = (mat[i][j] + tr[i][j]) / 2;
            skewsymm[i][j] = (mat[i][j] - tr[i][j]) / 2;
        }
    }
    
    printf("Original Matrix:<br>");
    printMatrix(mat);
    
    printf("\nSymmetric Matrix:<br>");
    printMatrix(symm);
    
    printf("\nSkew Symmetric Matrix:<br>");
    printMatrix(skewsymm);
    
    return 0;
}

Original Matrix:
2.0 -2.0 -4.0 
-1.0 3.0 4.0 
1.0 -2.0 -3.0 

Symmetric Matrix:
2.0 -1.5 -1.5 
-1.5 3.0 1.0 
-1.5 1.0 -3.0 

Skew Symmetric Matrix:
0.0 -0.5 -2.5 
0.5 0.0 3.0 
2.5 -3.0 0.0 

Key Points

• The diagonal elements of a skew-symmetric matrix are always zero
• The decomposition is unique for any square matrix
• The sum of the symmetric and skew-symmetric parts equals the original matrix

Conclusion

Any square matrix can be uniquely decomposed into symmetric and skew-symmetric components using the formulas shown above. This decomposition is fundamental in linear algebra and matrix theory.