We define the operators $$\mathcal{S}: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto NX - XN \end{cases} \quad \text{and} \quad \mathcal{S}^*: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto {}^t N X - X {}^t N \end{cases}$$ Show that the kernel of $\mathcal{S}$ is the set of real Toeplitz matrices that are lower triangular. We admit that the kernel of $\mathcal{S}^*$ is the set of real Toeplitz matrices that are upper triangular.
We define the operators
$$\mathcal{S}: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto NX - XN \end{cases} \quad \text{and} \quad \mathcal{S}^*: \begin{cases} \mathcal{M}_n(\mathbb{R}) \rightarrow \mathcal{M}_n(\mathbb{R}) \\ X \mapsto {}^t N X - X {}^t N \end{cases}$$
Show that the kernel of $\mathcal{S}$ is the set of real Toeplitz matrices that are lower triangular. We admit that the kernel of $\mathcal{S}^*$ is the set of real Toeplitz matrices that are upper triangular.