Antisymmetric matrix

Antisymmetric matrix

Mirror 1

An antisymmetric matrix is ​​a square matrix where the elements outside the main diagonal are symmetrically equal but those below the main diagonal carry a negative sign.

In other words, an antisymmetric matrix is ​​a matrix that has the same number of rows (n) and columns (m) and the elements on both sides of the main diagonal are complementary.

Since the elements above and below the main diagonal are offset, the elements on the main diagonal are zeros.

Recommended article: non-symmetric matrix and symmetric matrix.

Characteristics of the antisymmetric matrix

The characteristics of an antisymmetric matrix are:

  • Square matrix.
  • Symmetric matrix + negative sign (-) in the elements below the main diagonal.
  • Elements of the main diagonal are zeros (0).

Antisymmetric matrix

Given a square matrix AS ,

Antisymmetric Matrix 1
Antisymmetric matrix

We can see how the same elements appear on both sides of the main diagonal, but with the particularity that the elements below the main diagonal have a negative sign in front. Also, the main diagonal is made up of zeros.

The antisymmetric matrix and mirrors

In the same way as the symmetric matrix, the antisymmetric matrix can also be understood through the example of the mirror.

Mirror 1
Mirror

If we look at ourselves in the mirror and raise our right arm, we will see that the person in the mirror raises their left arm. In other words, the movement of the mirror complements ours and, therefore, the sum of both would result in zero.

We can express the above idea as follows and deduce:

(Raise the right hand) (Raise the left hand) = 0

(Raise the right hand) = (Raise the left hand)

The main diagonal acts as a mirror and we see opposing elements on both sides of the main diagonal. The neutral function (=) maps to the main diagonal.

Property

  • The transposed matrix of an antisymmetric matrix is ​​equal to the antisymmetric matrix multiplied by (-1).

In other words, it would be like adding a negative sign in front of the antisymmetric matrix.

Mathematically,

Antisymmetric Matrix Property 1
Property of the antisymmetric matrix

We can see that with both procedures we arrive at the same result: making the matrix transposed or multiplying by (-1) the antisymmetric matrix.

Non-symmetric matrix vs Antisymmetric matrix vs Symmetric matrix

The example of the mirror in the case of the symmetric matrix is ​​enough that it reflects the same movement, that is, if we raise an arm, we can see a raised arm but it is not necessary to specify what it is. In the case of the antisymmetric matrix, we need to check which arm we see in the mirror and determine if it is an antisymmetric matrix.

If we raise an arm and in the mirror we see that …

  • The same arm is raised, from the point of view of the person in the mirror, then it is a symmetrical matrix.
  • The opposite arm is raised, from the point of view of the person in the mirror, then it is an antisymmetric matrix.
  • If no arm is raised or more than one is raised, from the point of view of the person in the mirror, then it is a non-symmetric matrix.

Main diagonal

  • Matrix division
  • Cholesky decomposition
  • Attached matrix