Extended Chebyshev systems for the expansions of exp (At)

Resumen

Consider a matrix $A$ of order $n\times n$ having real entries (i.e., $A\in\mathbb{R}^{n\times n}$). The degree of its minimal polynomial is $\mu$. It is proved that the identity $exp(At)=\sum^{\rho-1}_{k=0}\alpha_k(t)A^k$ stands for sets ${\alpha_u(t):u=0,1,\cdots,\rho-1}$ of functions of real variable defined in any real interval $I$. These sets are unique for each integer $\rho\geq\mu$ and can be determined from a system of linear equations. In addition, these sets are always Chebyshev systems on a real interval $(\gamma,\gamma +\pi/\omega)$, with $\omega = max_{1\leq k\leq \sigma} (\Im(\lambda_k)), \lambda_k(k=1,2,\cdots,\sigma)$ being the eigenvalues of $A$, and any $\gamma\in\mathbb{R}$. These results generalize a weaker parallel known result which stands for the set of minimum cardinal (i.e., for $\rho=\mu$). The generalizations obtained lead to important consequences when solving some algebraic problems in control theory.