PhD Seminar • Symbolic Computation • Applications of the Smith Massager of a Nonsingular Integer Matrix

Friday, September 10, 2021 10:00 am - 10:00 am EDT (GMT -04:00)

Please note: This PhD seminar will be given online.

Stavros Birmpilis, PhD candidate
David R. Cheriton School of Computer Science

Supervisors: Professors George Labahn, Arne Storjohann

Given a nonsingular integer matrix A ∈ Zn×n with Smith normal form S = diag(s1, . . . , sn), we define a matrix M ∈ Zn×n to be a Smith massager for A. We use the notation cmod S to show that an equivalence is taken column modulo the diagonal entries in S. Matrix M satisfies (i) that AM ≡ 0 cmod S, namely, the matrix AMS−1 is integral, and (ii) that there exists a matrix W ∈ Zn×n such that WM ≡ In cmod S, namely, the Smith massager is “unimodular” up to equivalence column modulo S. We obtain the Smith massager from an algorithm that computes the Smith form of A. We show that M serves as a useful object for tackling other problems in integer linear algebra like computing the Smith multiplier matrices for A or representing the fractional part of the adjoint of A.

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