A non-symmetric matrix is a non-square matrix where the elements of the transposed matrix are in different positions than the elements of the original matrix.
In other words, the non-symmetric matrix is a matrix where the number of rows (n) is different than the number of columns (m) and the transpose of the matrix is different from the original matrix.
It is important not to confuse non-symmetric matrices with antisymmetric matrices since they are very different concepts and refer to different elements within the matrix.
For a matrix to be symmetric, it must be a square matrix and it must be equal to its transposed matrix. In other words, that the number of rows (n) is equal to the number of columns (m) and that the elements of the matrix do not change once the rows have been changed by the columns.
Mathematically the concept of symmetry means that applying the transpose operation, the elements of the matrix will not change.
The symmetric matrix and mirrors
We will better understand the concept of a non-symmetric matrix if we think about the effect that a mirror produces.
If we look in the mirror we will see our face reflected; if we raise a hand, a hand will also rise in the mirror. In the same way that if we make any gesture, the same reflected gesture will appear.
Well, the same happens with the main diagonal of a symmetric matrix. Items below or above the main diagonal will be the same. That is, the main diagonal of a symmetric matrix acts as a mirror of the elements around it.
Given a symmetric matrix S ,
The transposed matrix S would have the following form:
For more information on its mathematical properties, consult the article on symmetric matrix.
The non-symmetric matrix and mirrors
In the case of the non-symmetric matrix, it is as if the mirror was broken.
And when a mirror is broken it does not reflect well the elements in front of it. We can raise the right hand and see that four hands are raised or none are raised.
So, applying the same logic, the non-symmetric matrix is about not having the same elements above or below the main diagonal and also that they are not equal.
In this matrix we cannot find the main diagonal and, therefore, there is no symmetry in the number of elements. Furthermore, if we transpose the previous matrix we will see that it does not retain its original state.
The transposed NS matrix would have the following form:
When we come across the concept of a non-symmetric matrix, we only have to think about the symmetric matrix and put a negation in front of its characteristics. That is, a non-symmetric matrix will be such that it satisfies:
- Non- square matrix.
- Transposed matrix not equal to original matrix.
It may seem easy to remember what a non-symmetric matrix is, but when we work with antisymmetric matrices we sometimes confuse the concepts.