Certain matrix congruences (modp n)

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A canonical form is developed for matrices (mod pn). This form is used to enumerate the incongruent solutions of the matrix congruences

$$AX \equiv B, UA + BV \equiv C, X'A + A'X \equiv B, and X'A - A'X \equiv B (\bmod p^n )$$

. Solvability conditions are also provided in each case.


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    L. E. Fuller,A canonical set of matrices over a principal ideal ring modulo m, Can. J. Math.,7 (1955), pp. 54–58.

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    J. H. Hodges,Some matrix equations over a finite field, Annali di Matematica,44 (1957), pp. 245–250.

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    J. H. Hodges,The matric equation AX = B in a finite field, American Mathematical Monthly,63 (1956), pp. 243–244.

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    B. R. McDonald,Enumeration of canonical matrices under row equivalence over a principal ideal domain modulo p n, Duke Mathematical Journal,38 (1970), pp. 393–402.

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Entrata in Redazione il 15 settembre 1976.

This paper represents a portion of my doctoral thesis under Professor JohnH. Hodges to whom I am sincerely grateful.

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Donovan, T.P. Certain matrix congruences (modp n). Annali di Matematica 115, 193–214 (1977) doi:10.1007/BF02414717

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  • Canonical Form
  • Solvability Condition
  • Matrix Congruence
  • Incongruent Solution