skew-symmetric matrix

Definition:
Let $A$ be an square matrix oforder $n$ with real entries $(a_{ij})$.The matrix $A$ is skew-symmetric if $a_{ij}=-a_{ji}$ for all $1\le i\le n,1\le j\le n$.

The main diagonal entries are zero because$a_{i,i}=-a_{i,i}$ implies $a_{i,i}=0$.

One can see skew-symmetric matrices as aspecial case of complex skew-Hermitian matrices. Thus,all properties of skew-Hermitian matrices also holdfor skew-symmetric matrices.

Properties:

1. The matrix $A$ is skew-symmetric if and only if$A^t=-A$, where $A^t$ is the matrix transpose

2. For the trace operator, we have that$\mathrm{tr}(A)=\mathrm{tr}(A^t)$.Combining this with property (1), it followsthat $\mathrm{tr}(A)=0$ for a skew-symmetric matrix $A$.

3. Skew-symmetric matrices form a vector space: If $A$ and $B$are skew-symmetric and $\alpha ,\beta \in ℝ$, then$\alpha A+\beta B$ is also skew-symmetric.

4. Suppose $A$ is a skew-symmetric matrix and $B$ is a matrix ofsame order as $A$. Then $B^{t}AB$ is skew-symmetric.

5. All eigenvalues of skew-symmetric matrices arepurely imaginary or zero. This result is proven on the pagefor skew-Hermitian matrices.

6. According to Jacobi’s Theorem, the determinant of askew-symmetric matrix of odd order is zero.

Examples: