What is Skew Symmetric Matrix? Properties & Examples

Skew symmetric matrix is a square matrix Q=[xij] in which (i, j)th element is negative of the (j, i)th element i.e. xij = -xji for all values of i and j. In other words, a square matrix (Q) which is equal to negative of its transpose is known as skew-symmetric matrix i.e. QT = -Q.

Skew symmetric matrix examples

The example of a skew-symmetric matrix of order 2 (or matrix size is 2 x 2) is given as

    \[C=\begin{bmatrix} 0 &2 \\ -2& 0 \end{bmatrix}\]

The example of a skew-symmetric matrix of order 3 (or matrix size is 3 x 3) is given as

    \[D=\begin{bmatrix} 0 &8 &2 \\ -8& 0 &5 \\ -2& -5 & 0 \end{bmatrix}\]

Skew symmetric matrix properties

1. All the diagonal elements in a skew-symmetric matrix are always zero.

2. If P be a skew-symmetric matrix then PT = -P.

Let B be any matrix then

1. \frac{B-B^T}{2}

is always a skew-symmetric matrix.

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