New possibilities of Gramian computation, by means of canonical transformations into
diagonal, controllable, and observable canonical forms, are shown. Using such a technique, the
Gramian matrices can be represented as products of the Hadamard matrices of multipliers and
the matrices of the transformed right-hand sides of Lyapunov equations. It is shown that these
multiplier matrices are invariant under various canonical transformations of linear continuous
systems. The modal Lyapunov equations for continuous SISO LTI systems in diagonal form are
obtained, and their new solutions based on Hadamard decomposition are proposed. New algorithms
for the element-by-element computation of Gramian matrices for stable, continuous MIMO LTI
systems are developed. New algorithms for the computation of controllability Gramians in the
form of Xiao matrices are developed for continuous SISO LTI systems, given by the equations of
state in the controllable and observable canonical forms. The application of transformations to the
canonical forms of controllability and observability allowed us to simplify the formulas of the spectral
decompositions of the Gramians. In this paper, new spectral expansions in the form of Hadamard
products for solutions to the algebraic and differential Sylvester equations of MIMO LTI systems
are obtained, including spectral expansions of the finite and infinite cross-Gramians of continuous
MIMO LTI systems. Recommendations on the use of the obtained results are given.