Abstract
A canonical form for a reduced matrix of order 3 with one characteristic root and with some zero subdiagonal elements is constructed. Thus, the problem of classification with respect to semiscalar equivalence of a selected set of polynomial matrices is solved.
Highlights
Let a matrix F(x) ∈ M(n, C[x]) have a unit first invariant factor and only one characteristic root
If i = 1 and ss.e.t.-I is applied to matrix A(x), the elements a1(x) ≠ 0, a2(x), a3(x) ≡ 0 and b1(x), b2(x), b3(x) of the matrices A(x) and B(x) satisfy the following congruences: b1 (x) − a1 (x) (1 − s12b1 (x)) ≡ 0, (11)
From ss.e. of the matrices A(x) and B(x) it follows that their elements satisfy the congruences: b1 (x) + s13a3 (x) b1 (x) ≡ 0, (17)
Summary
Let a matrix F(x) ∈ M(n, C[x]) have a unit first invariant factor and only one (without taking into account multiplicity) characteristic root. Denote by symbol +∞ the junior degree of the polynomial f(x) ≡ 0 If both elements a1(x), a2(x) of the matrix A(x) are nonzero, we may take their junior coefficients to be identity elements. We may take the junior coefficients of the nonzero subdiagonal elements of the matrix A(x) to be one. Such matrix A(x) in [3] is called the reduced matrix. In this paper we consider the case, when some of the elements a1(x), a2(x), a3(x) of the matrix A(x) are equal to zero and at least one of them is different from zero. We add that the work close to [1, 2] is [10, 11]
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