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"I liked the algebraic way of looking at things.
I’m additionally fascinated when the algebraic method is applied to infinite objects”.

Irwing Kaplansky

# Matrices are reduced to canonical diagonal form by an invertible Toeplitz matrices

Vasylyna Bokhonko

Department of Algebra and Logic

Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv

Abstract:

Under a ring we introduce a commutative ring R with 1≠0. A Bezout ring is the ring in which every finitely generated ideal is principal.

Definition 1. A commutative Bezout ring R we call a Toeplitz ring if for any a,b ϵ R there exists T − an invertible Toeplitz matrix such that (a,b)T=(d,0).

Теорема 1. Let R is a commutative Hermite ring of stable range one. Then R is the Toeplitz ring.

Definition 2. A ring R called ring of unit square stable range one if from condition aR+bR=R is following that there exists a unit element t such that a2+bt − the unit element of the ring R.

Theorem 2. Let R is a commutative Hermite ring unit square stable range one. Then any matrix of the second order over R is reduced to diagonal form by the invertible Toeplitz matrices.

Theorem 3. Let R a commutative Bezout ring of stable range one. Then:
1) some unimodular row (column) of length 2 over R is reduced to the invertible Toeplitz matrix;
2) some matrix of the second order over R is reduced to canonical diagonal form by the right and left multiplication invertible Toeplitz matrices;
3) some invertible matrix of the second order over ring R is decomposed in product the invertible Toeplitz matrices.

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Department of Algebra and Logic
Faculty of Mechanics and Mathematics
Ivan Franko National University of L'viv
1 Universytetska Str., 79000 Lviv, Ukraine
Tel: (+380 322) 394 172
E-mail: oromaniv at franko.lviv.ua