skew-symmetric matrix

Let A be an square matrixMathworldPlanetmath of order n with real entries (aij). The matrix A is skew-symmetric if aij=-aji for all 1in,1jn.


The main diagonal entries are zero because ai,i=-ai,i implies ai,i=0.

One can see skew-symmetric matrices as a special case of complex skew-Hermitian matrices. Thus, all properties of skew-Hermitian matrices also hold for skew-symmetric matrices.


  1. 1.

    The matrix A is skew-symmetric if and only if At=-A, where At is the matrix transpose

  2. 2.

    For the trace operator, we have that tr(A)=tr(At). Combining this with property (1), it follows that tr(A)=0 for a skew-symmetric matrix A.

  3. 3.

    Skew-symmetric matrices form a vector spaceMathworldPlanetmath: If A and B are skew-symmetric and α,β, then αA+βB is also skew-symmetric.

  4. 4.

    Suppose A is a skew-symmetric matrix and B is a matrix of same order as A. Then BtAB is skew-symmetric.

  5. 5.

    All eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of skew-symmetric matrices are purely imaginary or zero. This result is proven on the page for skew-Hermitian matrices.

  6. 6.

    According to Jacobi’s Theorem, the determinantMathworldPlanetmath of a skew-symmetric matrix of odd order is zero.


  • (0b-b0)

  • (0bc-b0e-c-e0)

Title skew-symmetric matrix
Canonical name SkewsymmetricMatrix
Date of creation 2013-03-22 12:01:05
Last modified on 2013-03-22 12:01:05
Owner Daume (40)
Last modified by Daume (40)
Numerical id 10
Author Daume (40)
Entry type Definition
Classification msc 15-00
Related topic SelfDual
Related topic AntiSymmetric
Related topic SkewHermitianMatrix
Related topic AntisymmetricMapping