What is skew Symmetric

Let A be a square matrix if the transpose of this matrix is equal to the negative of that matrix then it is called skew symmetric matrix. In linear algebra skew-symmetric matrix defines as the transpose of the matrix that is equal to its negative matrix. The skew-symmetric matrix must be a square matrix. In the skew-symmetric matrix, the main diagonals entries are zero.

skew Symmetric Matrix Example
skew Symmetric Matrix

We also define skew-symmetric matrix as A^t = -A. Let A be a square matrix of order n by n  we denote the entries as aij in skew-symmetric when we take the transpose of matrix A then these entries will be changed (aij)^t = -(aij) where i denote the number of rows and j shows the number of columns.

Example of skew-symmetric matrix

No 1.

Skew-symmetric matrix of A:

A=\begin{bmatrix}0&-2&3\\2&0&4\\-3&-4&0\end{bmatrix},

Order of this matrix is 3 by 3.

A^t=\begin{bmatrix}0&-2&3\\2&0&4\\-3&-4&0\end{bmatrix}^t \Rightarrow A^t=\begin{bmatrix}0&2&-3\\-2&0&-4\\3&4&0\end{bmatrix} \Rightarrow A^t=-\begin{bmatrix}0&-2&3\\2&0&4\\-3&-4&0\end{bmatrix}

A^t= –A

hence this is skew symmetric matrix.

No 2.

skew-symmetric matrix of B:

B=\begin{bmatrix}0&-2\\2&0\end{bmatrix}

Order of this matrix is 2 by 2.

B^t=\begin{bmatrix}0&-2\\2&0\end{bmatrix}^t \Rightarrow B^t=\begin{bmatrix}0&2\\-2&0\end{bmatrix} \Rightarrow B^t=-\begin{bmatrix}0&-2\\2&0\end{bmatrix}

B^t= –B

so this matrix is skew symmetric matrix.

Properties of a skew-symmetric matrix:

  • When we have scalar product of two skew symmetric matrices then the resultant matrix also skew symmetric matrix.
  • Eigen value of real skew symmetric matrices are zero however non zero symmetric matrices have non zero eigen values.
  • when we add two skew symmetric matrices then the matrix that we obtain is skew symmetric matrix.
  • the trace of skew symmetric matrices are zero because the trace is the sum of main diagonals and the main diagonals of the skew symmetric matrices are zero.
  • the determinant of skew symmetric matrices are having odd order that is zero.
  • when we added identity matrix in any skew symmetric matrix then the matrix that we obtain is invertible.
  • skew symmetric matrix is diagonalizable.
  • The rank of skew symmetric matrices are always even.
Here the question is arises that why the diagonals elements of skew symmetric matrix are always zero? The diagonals elements of skew symmetric matrix are always zero because when the main diagonals are non zero then the transpose and negative of that matrix does not equals. let we explain this in detail.

Let A be a square matrix

A=\begin{bmatrix}1&2\\-2&4\end{bmatrix}

first, we take the transpose of this matrix. for transpose the first-row change in the first column and second-row change in the second column.

A^t=\begin{bmatrix}1&-2\\2&4\end{bmatrix}

Now we take negative of this matrix for this we change the negative value into positive and positive values in negative.

-A=\begin{bmatrix}-1&-2\\2&-4\end{bmatrix}

A^t not is equal to A

here the property of skew-symmetric does not hold two matrices are equal if and only if when their order is same and all elements in matrices are same and here both matrices are different so the property of skew-symmetric that the transpose of a matrix is equal to its negative matrix is not hold

Note skew-symmetric matrix

since the skew-symmetric matrix is singular therefore the inverse of the skew-symmetric matrix does not exist.

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