Abstract
For the selected class of polynomial matrices of order three with one characteristic root with respect to the transformation of semiscalar equivalence, special triangular forms are established. The theorems of their uniqueness are proved. This gives reason to consider such canonical forms.
Highlights
In [1], it is proved that the matrix F(x) ∈ M(n, C[x]) of full rank by means of transformation: F(x) ⟶ PF(x)Q(x) G(x), (1)where P ∈ GL(n, C) and Q(x) ∈ GL(n, C[x]) can be reduced to the lower triangular form with invariant factors on the principal diagonal
Subdiagonal elements in a matrix of this form are ambiguously defined. e matrices F(x), G(x) which are related by the transformation (1) are called semiscalarly equivalent [1]
Conclusion e matrices B(x), whose existence is established in eorems 1 and 2, can be considered canonical in the class of semiscalarly equivalent matrices. e method of their construction follows from the proof of the first parts of these theorems. is completes the study of semiscalar equivalence of third-order polynomial matrices with one characteristic root, started in the previous works of the author
Summary
In [1], it is proved that the matrix F(x) ∈ M(n, C[x]) of full rank by means of transformation: F(x) ⟶ PF(x)Q(x) G(x),. Where P ∈ GL(n, C) and Q(x) ∈ GL(n, C[x]) can be reduced to the lower triangular form with invariant factors on the principal diagonal. Subdiagonal elements in a matrix of this form are ambiguously defined. E matrices F(x), G(x) which are related by the transformation (1) are called semiscalarly equivalent [1]. E resulting matrix of a simplified triangular form is called a reduced. The invariance of the location of zero subdiagonal elements is proved. In [3], the reduced matrix, if there are some zero elements under its principal diagonal, by means of transformations of the form (1) (i.e., by means of semiscalarly equivalent transformations) is reduced to such matrices, which are uniquely defined. Is article introduces canonical forms for reduced matrices with all nonzero subdiagonal elements In [3], the reduced matrix, if there are some zero elements under its principal diagonal, by means of transformations of the form (1) (i.e., by means of semiscalarly equivalent transformations) is reduced to such matrices, which are uniquely defined. is gives grounds to consider the obtained matrices canonical for the selected class of matrices. is article introduces canonical forms for reduced matrices with all nonzero subdiagonal elements
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