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

Symmetric Matrix − A matrix whose transpose is equal to the matrix itself. Then it is called a symmetric matrix.

Skew-symmetric matrix − A matrix whose transpose is equal to the negative of the matrix, then it is called a skew-symmetric matrix.

The sum of symmetric and skew-symmetric matrix is a square matrix. To find these matrices as the sum we have this formula.

Let A be a square matrix. then,

A = (½)*(A + A`)+ (½ )*(A - A`),

A` is the transpose of the matrix.

(½ )(A+ A`) is symmetric matrix.

(½ )(A - A`) is a skew-symmetric matrix.

Example

#include <bits/stdc++.h>
using namespace std;
#define N 3
void printMatrix(float mat[N][N]) {
   for (int i = 0; i < N; i++) {
      for (int j = 0; j < N; j++)
         cout << mat[i][j] << " ";
         cout << endl;
   }
}
int main() {
   float mat[N][N] = { { 2, -2, -4 },
   { -1, 3, 4 },
   { 1, -2, -3 } };
   float tr[N][N];
   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];
   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;
      }
   }
   cout << "Symmetric matrix-" << endl;
   printMatrix(symm);
   cout << "Skew Symmetric matrix-" << endl;
   printMatrix(skewsymm);
   return 0;
}

Output

Symmetric matrix -
2 -1.5 -1.5
-1.5 3 1
-1.5 1 -3
Skew Symmetric matrix -
0 -0.5 -2.5
0.5 0 3
2.5 -3 0