### A direct transformation between two fundamental matrix canonical forms

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In this paper we construct analytic-numerical solutions for initial-boundary value systems related to the equation ${u}_{t}-A{u}_{xx}-Bu=0$, where $B$ is an arbitrary square complex matrix and $A$ ia s matrix such that the real part of the eigenvalues of the matrix $\frac{1}{2}(A+{A}^{H})$ is positive. Given an admissible error $\epsilon $ and a finite domain $G$, and analytic-numerical solution whose error is uniformly upper bounded by $\epsilon $ in $G$, is constructed.