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Keywords

Linear transformation, Dimension formulas, Jordan canonical form

Abstract

Let α be a linear transformation of the m × n-dimensional vector space M_{m×n}(C) over the complex field C such that α(X) = AX −XB, where A and B are m×m and n×n complex matrices, respectively. In this paper, the dimension formulas for the kernels of the linear transformations α^2 and α^3 are given, which generalizes the work of Gracia in [J.M. Gracia. Dimension of the solution spaces of the matrix equations [A, [A, X]] = 0 and [A[A, [A, X]]] = 0. Linear and Multilinear Algebra, 9:195–200, 1980.].

abs_vol30_pp243-252.pdf (33 kB)
Abstract

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