Antisymmetric matrix

An antisymmetric (or skew-symmetric) matrix is a matrix <math>A</math> such that <math>A^T = -A</math>. In other words, a matrix is antisymmetric if it is equal to its negative transpose.

For example, the matrix

<math>A = \begin{bmatrix}

   0  & 1 & 2  \\
   -1 & 0 & -6 \\
   -2 & 6 & 0
   \end{bmatrix}</math>

is anti-symmetric, because

<math>-A = \begin{bmatrix}

   0  & -1 & -2  \\
   1 & 0 & 6 \\
   2 & -6 & 0
   \end{bmatrix}
   = A^T</math>.

Properties

• If you add two antisymmetric matrices, the result is another antisymmetric matrix.
• If you multiply an antisymmetric matrix by a constant, the result is another antisymmetric matrix.
• All of the diagonal entries of an antisymmetric matrix are 0.

Applications

For an electromagnetic field, the curvature form is an antisymmetric matrix whose elements are the electric field and magnetic field: the electromagnetic tensor.

Related pages

• Symmetric matrix