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<p>We study the problem of divisibility with remainder of polynomial matrices over an arbitrary field <i>F</i> and establish the conditions under which, for a pair of polynomial matrices <i>A</i>(λ) and <i>B</i>(λ) over the field <i>F</i> , there exists a unique pair of polynomial matrices <i>P</i>(λ) and <i>Q</i>(λ) over <i>F</i> such that <i>B</i>(λ) = <i>A</i>(λ)<i>P</i>(λ) +<i>Q</i>(λ) . We also discuss the application of the obtained results to finding minimal solutions of the Sylvester-type matrix equation. It is proved that nonsingular polynomial matrices <i>A</i>(λ) and <i>B</i>(λ) over <i>F</i> have relatively prime determinants if and only if the matrix equation <i>A</i>(λ)<i>X</i>(λ) + <i>Y</i> (λ)<i>B</i>(λ) = <i>C</i>(λ) has a unique minimal solution for any nonzero matrix <i>C</i>(λ).</p>

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On the Divisibility with Remainder of Polynomial Matrices Over an Arbitrary Field

  • 期刊论文

  • 发表日期：

    2025年7月28日

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  • V. M. Prokip,
  • O. M. Mel’nyk,
  • R. V. Kolyada

摘要

We study the problem of divisibility with remainder of polynomial matrices over an arbitrary field F and establish the conditions under which, for a pair of polynomial matrices A(λ) and B(λ) over the field F , there exists a unique pair of polynomial matrices P(λ) and Q(λ) over F such that B(λ) = A(λ)P(λ) +Q(λ) . We also discuss the application of the obtained results to finding minimal solutions of the Sylvester-type matrix equation. It is proved that nonsingular polynomial matrices A(λ) and B(λ) over F have relatively prime determinants if and only if the matrix equation A(λ)X(λ) + Y (λ)B(λ) = C(λ) has a unique minimal solution for any nonzero matrix C(λ).