Suppose \(\mathbf{A}\) is an \(m\)-by-\(n\) matrix with row vectors \(\vec{\alpha}_1,\ldots,\vec{\alpha}_m\) and \(\mathbf{B}\) is an \(n\)-by-\(p\) matrix with column vectors \(\vec{b}_1,\ldots,\vec{b}_p.\)

A square matrix \(\mathbf{A}=(A_{ij})_{1\leqslant i,j\leqslant n} \in M_{n,n}(\mathbb{K})\) is said to be symmetric if \(A_{ij}=A_{ji}\) for all \(1\leqslant i,j\leqslant n.\) Show that if \(\mathbf{A},\mathbf{B}\) are symmetric matrices such that \(\mathbf{A}\mathbf{B}\) is symmetric as well, then they satisfy \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}.\)

Let \(n \in\mathbb{N}\) and \(\mathbf{A}\in M_{n,n}(\mathbb{R})\) be a matrix such that \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}\) for all \(\mathbf{B}\in M_{n,n}(\mathbb{R}).\)

We consider the set \(\mathsf{P}_n(\mathbb{R})\) of polynomials of degree at most \(n \in \mathbb{N}\cup \{0\}\) with real coefficients in one real variable \(x,\) so that a function \(g \in \mathsf{P}_n(\mathbb{R})\) is of the form \(g(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+ a_1x+a_0\) for real coefficients \(a_n,a_{n-1},\ldots,a_1,a_0 \in \mathbb{R}.\) Notice that we have a bijective mapping \[\Psi : \mathbb{R}^{n+1} \to \mathsf{P}_n(\mathbb{R}),\qquad \begin{pmatrix} a_n \\ a_{n-1} \\ \vdots \\ a_1 \\ a_0 \end{pmatrix} \mapsto a_nx^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0.\] Find a matrix \(\mathbf{A}\in M_{n+1,n+1}(\mathbb{R})\) such that for all \(g \in \mathsf{P}_{n}(\mathbb{R})\) we have \[g^{\prime}=\left(\Psi\circ f_\mathbf{A}\circ \Psi^{-1}\right)(g),\] where the prime denotes the derivative by the variable \(x.\)

Hint: Try to understand the cases \(n=1,\) \(n=2\) and \(n=3\) first.

(\(\heartsuit\) – for the enthusiast). Find a function \(f : \mathbb{R}^2 \to \mathbb{R}^2\) which is \(1\)-homogeneous, but not additive.