Let $W_1$ be the subspace of $\mathcal{M}_{n \times n}$ that consists of all $n \times n$ skew-symmetric matrices with entries from $\mathbb{F}$, and let $W_2$ be the subspace of $\mathcal{M}_{n \times n}$ consisting of all symmetric $n \times n$ matrices. Prove that $\mathcal{M}_{n \times n}(\mathbb{F}) = W_1 \oplus W_2$.
I couldn't really figure out why the sum of an $n \times n$ symmetric matrix and $n \times n$ skew-symmetric matrix would form a $n \times n$ matrix (to satisfy the direct summand property $W_1 + W_2 = \mathcal{M}_{n \times n}(\mathbb{F})).$ Browsing online, I found that $$ M = \frac{1}{2}(M + M^{t}) + \frac{1}{2}(M-M^{t}),$$ where $\frac{1}{2}(M+M^{t}) \in W_2, \frac{1}{2}(M-M^t) \in W_1$, and $M \in \mathcal{M}_{n \times n}(\mathbb{F})$.
This reminds me of a formula on how odd and even functions may be be added together to form a generic function. $$f(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}.$$
However, to me it is disturbing to use unless I know how it was derived. If anyone could show me why a skew-symmetric matrix may be represented as $\frac{1}{2}(M-M^t)$ and why a symmetric matrix may be represented as $\frac{1}{2}(M+M^t)$ than I may sleep better tonight.
Thanks.
Best Answer
Rather than asking why a symmetric matrix may be represented as $\frac{1}{2}(M+M^t)$, you need to ask yourself the following three things:
After you have answered "yes" to the above three questions, you will have proven that $M_{n,n}(\mathbb{F})=W_1+W_2$. Then, you merely prove that $W_1\cap W_2=\{0\}$, and then you have $M_{n,n}(\mathbb{F})=W_1\oplus W_2$.
Ultimately, there are many matrices $M$ that all have the same symmetrization via the above formula. The reason those two formulas are used is that they make the above three questions all have the right answer.