[Math] Does the set of skew-symmetric n×n matrices form a vector space

linear algebramatrices

Does the set of skew-symmetric n×n matrices form a vector space with the usual matrix addition and scalar multiplication?

This is quite easy to prove if we take a specefic dimension like 2×2, but I am quite confused about poving it for all nxn square matricies.

Best Answer

Let $\operatorname{Mat}_n$ be the set of $n \times n$ matrices and $\operatorname{Skew}_n = \left\{ A \in \operatorname{Mat}_n \;\middle|\; A^T = - A \right\}$ be the set of $n \times n$ skew-symmetric matrices. The answer to your question is yes. The easiest way to see this is by showing that $\operatorname{Skew}_n$ is a vector subspace of the vector space $\operatorname{Mat}_n$. To do that, let we have to show that

  1. $0 \in \operatorname{Skew}_n$
  2. $A, B \in \operatorname{Skew}_n \implies A + B \in \operatorname{Skew}_n$
  3. $\alpha$ a scalar, $A \in \operatorname{Skew}_n \implies \alpha A \in \operatorname{Skew}_n$.

Here are the proofs:

  1. $0^T = 0 = -0$, so $0 \in \operatorname{Skew}_n$.
  2. $(A+B)^T = A^T + B^T = -A + (-B) = -(A+B)$
  3. $(\alpha A)^T = \alpha A^T = - \alpha A$.
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