A symmetric matrix is a matrix of order n with the same number of rows and columns where its transposed matrix is equal to the original matrix.
In other words, a symmetric matrix is a square matrix and is identical to the matrix after having swapped rows for columns and columns for rows.
Requirements
For any matrix to be a symmetric matrix, it must meet the following restrictions:
Given a symmetric matrix P of order n,
- Be a square matrix .
The number of rows (n) has to be the same as the number of columns (m). That is, the order of the matrix has to be n given that n = m.
- The original matrix has to be equal to its transposed matrix .
Demonstration:
Properties
- The adjoint matrix of a symmetric matrix is also a symmetric matrix.
Demonstration:
- The addition or subtraction of two symmetric matrices results in another symmetric matrix.
Demonstration:
Given two symmetric matrices P and T of order 3, we obtain another symmetric matrix S from the sum.
Why is it called a symmetric matrix?
The property of symmetry is given by the elements around the main diagonal. Since a square matrix is a symmetric matrix, it will always have the same number of elements above and below the main diagonal. These elements are the same symmetrically. That is, the main diagonal acts as a mirror.
Proof of symmetry and skewness of a matrix
Symmetric matrix
The letter d represents the elements of the main diagonal. The other letters represent any real number. We can see that the main diagonal acts like a mirror: it reflects the elements on both sides. In other words, when the elements on both sides of the diagonal are symmetrically equal, we say that the matrix P is a symmetric matrix.
Non-symmetric matrix
Matrix X is not a symmetric matrix since it is not a square matrix and its transposed matrix is different from the original matrix. In addition, it does not have a main diagonal either.
Non-symmetric matrix