Method of ${\mathcal{M}}_n$-Extension via Frobenius Companion Matrices

Frobenius companion matrices arise when we write an $n$-th order linear ordinary differential equation as a system of first order differential equations. These matrices and their transpose have very nice properties. By using the powers of these matrices we form a closed algebra under the matrix multiplication. Structure constants of this commuting algebra are the components of companion matrix. We use these matrices in our method of ${\mathcal{M}}_n$-extension of scalar integrable equations to produce new systems of integrable equations with recursion operators.