Let $\{a_n\}, {n\geq 1}$, be a sequence of real numbers satisfying $|a_n|\leq 1$ for all $n$. Define $$A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n),$$ for $n\geq 1$. Then find $\displaystyle\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ . I proceed in this way $$\lim_{n \rightarrow \inft...

Calculate for $ n \geq 2 $ and $ x, a_{0}, a_{1}, \ldots, a_{n-1} \in \mathbb{R} $ the determinant of the following matrix: $$\begin{bmatrix} {x} & {0} & {\cdots} & {\cdots} & {\cdots} & {0} & {a_{0}} \\ {-1} & {x} & {0} & {\cdots} & {\cdots} & {0} & {a_{1}} \\ {0} & {-1} & {x} & {0} & {...

I'm looking for interesting applications of companion matrices. I can also use the Frobenius Normal Form. I already covered the Cayley-Hamilton Theorem and the application to linearly recursive sequences and high-order scalar linear differential equations.

Let $K$ be an arbitrary field and let $A\in\mathbb{M}_n(K)$. Let $m(X)$ and $p(X)$ be the minimal polynomial of $A$ over $K$ and the characteristic polynomial of $A$ over $K$, respectively. According to the Wikipedia article about companion matrix, the following are equivalent: $~~\deg(m(X))=\...