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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
Updated on: 19-Aug-2019
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