I am working on trying to prove that $$\text{Mat}(n,\mathbb{R}) = \mathcal{S} \oplus \mathcal{T}$$ where $\mathcal{S}$ is the set of all skew-symmetric, real matrices of size n and $\mathcal{T}$ is the set of all upper-triangular real matrices of size n.
I have shown these are of course subspaces of $\text{Mat}(n,\mathbb{R})$, and I proceeded by showing that both:
$$\mathcal{S} \cap \mathcal{T} = 0$$
$$\text{Mat}(n,\mathbb{R}) = \mathcal{S} + \mathcal{T}$$
where the second is interpreted that any real matrix can be written as the sum of a skew-symmetric matrix with an upper triangular matrix.
The first statement was easy to prove – though I am not making huge progress with the second half. I know that any real matrix $A$ can be written as the sum of a symmetric matrix with a skew-symmetric matrix using the fact $A \pm A^T$ is symmetric/anti-symmetric, but this of course isn't helping much for triangular matrices.
Any tips on how to proceed with this would be great.
Thanks!!
Best Answer
Hint: Let $A$ be a matrix in $M_n(\mathbb{R})$. We want to write it as $A=S+T$ where $S\in\mathcal{S}$ and $T\in\mathcal{T}$. Note that we don't have many options on how to define $S$ and $T$. Since the entries of $T$ below the main diagonal will all be zeros, we conclude that the entries of $S$ below the diagonal must be equal to the corresponding entries of $A$. So define $S_{ij}=A_{ij}$ whenever $i>j$. And since $S$ should be skew-symmetric, this tells us exactly how we must define all its other entries. And once we have $S$, this also tells us how we must define $T$. Can you finish from here?