Geodesic
Generalizations of a well-known result in matrix theory
Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 29-31

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Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F. 
Thompson, R. C. Generalizations of a well-known result in matrix theory. Glasgow mathematical journal, Tome 7 (1965) no. 1, pp. 29-31. doi: 10.1017/S2040618500035127
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[1] 1.Perlis, S., The theory of matrices (Cambridge, Mass., 1952). Google Scholar

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