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A left (right) block modal matrix is constructed to decompose a class of linear time-invariant MIMO state equations in arbitrary co-ordinates into a block diagonal canonical form which contains left (right) solvents of a characteristic polynomial matrix. The applications of block modal matrices to block partial fraction expansions, analytical time-response solutions, and model reductions of high-degree matrix fraction descriptions and high-order state equations are examined. Also, the relationship between a left (right) solvent of a matrix polynomial and the corresponding right (left)solvent is explored. The established block canonical forms in the time domain, and developed algebraic theories, provide additional mathematical tools for the analysis and synthesis of a class of MIMO control systems and matrix functions.