Smith Normal Form over Local Rings and Related Problems by Mustafa Elsheikh A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Computer Science Waterloo, Ontario, Canada, 2017 c Mustafa Elsheikh 2017 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner Jean-Guillaume Dumas Professeur, Math´ematiques Appliqu´ees Universit´eGrenoble Alpes, France Supervisor Mark Giesbrecht Professor, David R. Cheriton School of Computer Science University of Waterloo, Canada Internal Member George Labahn Professor, David R. Cheriton School of Computer Science University of Waterloo, Canada Internal Member Arne Storjohann Associate Professor, David R. Cheriton School of Computer Science University of Waterloo, Canada Internal-external Member Kevin Hare Professor, Department of Pure Mathematics University of Waterloo, Canada ii Author's Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions I am the sole author of Chapters 1, 4, and 7. Chapter 2 is based on an article co- authored with Mark Giesbrecht, B. David Saunders, and Andy Novocin. Chapter 3 is based on collaboration with Mark Giesbrecht, and B. David Saunders. Chapter 5 is based on an article co-authored with Mark Giesbrecht. Chapter 6 is based on collaboration with Mark Giesbrecht, and Andy Novocin. iv Abstract The Smith normal form is a diagonalization of matrices with many applications in diophantine analysis, graph theory, system control theory, simplicial homology, and more recently, in topological analysis of big data. Efficient computation of Smith normal form is a well-studied area for matrices with integer and polynomial entries. Existing successful algorithms typically rely on elimination for dense matrices and iterative Krylov space methods for sparse matrices. Our interest lies in computing Smith normal form for sparse matrices over local rings, where traditional iterative methods face challenges due to the lack of unique minimal poly- nomials. We explore different approaches to tackling this problem for two local rings: the integers modulo a prime power, and the polynomials modulo a power of an irreducible polynomial. Over local polynomial rings, we find success in linearization into larger di- mension matrices over the base field. Effectively we transform the problem of computing the Smith normal form into a small number of rank problems over the base field. The latter problem has existing efficient algorithms for sparse and dense matrices. The problem is harder over local integer rings. We take the approach of hybrid sparse- dense algorithms. We also tackle a restricted version of the problem where we detect only the first non-trivial invariant factor. We also give an algorithm to find the first few invariant factors using iterative rank-1 updates. This method becomes dense when applied to finding all the invariant factors. We digress slightly into the related problem of preconditioning. We show that linear- time preconditioners are suitable for computing Smith normal form, and computing nullspace samples. For the latter problem we design an algorithm for computing uniform samples from the nullspace. On a separate track, we focus on the properties of the Smith normal form decomposition. We relate the invariant factors to the eigenvalues. Our ultimate goal is to extend the applications of numerical algorithms for computing eigenvalues to computing the invariant factors of symbolic matrices. v Acknowledgements I would like to thank Mark Giesbrecht for being a mentor, a collaborator, and a good friend. I would like to thank my family for their constant love and support. Thank you Inas. vi Table of Contents 1 Introduction1 1.1 Smith Normal Form..............................1 1.2 The Black-Box Model..............................4 1.3 Probabilistic Algorithms............................9 1.4 Thesis Outline.................................. 10 2 Sparse Smith Normal Form 12 2.1 Introduction................................... 12 2.2 Previous Work................................. 14 2.2.1 Finding the Last Invariant Factor................... 14 2.2.2 The Valence Method.......................... 15 2.2.3 Using the Characteristic Polynomial................. 17 2.3 Linearization of Polynomial Matrices..................... 18 2.3.1 A Black-Box for the Embedding.................... 22 2.3.2 The Algorithm............................. 22 2.4 Smith Normal Form over Z=peZ ....................... 24 2.4.1 Nullspace Method............................ 24 2.4.2 Probabilistic Dimension Reduction.................. 25 vii 2.4.3 Working with Small Primes...................... 28 2.4.4 The Algorithm............................. 28 2.5 Detecting Non-Trivial Smith Normal Form.................. 30 3 Preconditioning 34 3.1 Introduction................................... 34 3.2 Nullspace Sampling............................... 36 3.2.1 Transpose Preconditioners....................... 37 3.2.2 Avoiding Extension Fields....................... 41 3.3 Smith Normal Form.............................. 43 3.3.1 Preconditioning and Determinantal Divisors............. 44 3.4 Conclusion.................................... 48 4 Rank Reduction 49 4.1 Introduction................................... 49 4.2 Invariant Factors and Rank-1 Updates.................... 52 4.3 Block Reduction................................ 59 4.4 Nullspace Sampling............................... 64 5 The Eigenvalues and the Invariant Factors 68 5.1 Introduction................................... 68 5.1.1 The p-adic Correspondence...................... 72 5.1.2 Previous Work............................. 72 5.2 Establishing p-Correspondence......................... 74 5.3 Density of p-Characterized Matrices...................... 78 5.3.1 Density at Large Primes........................ 82 5.4 Density at Small Primes............................ 89 viii 6 Ranks of Remainder Matrices 91 6.1 Introduction................................... 91 6.2 Quotient and Remainder Matrices....................... 92 6.2.1 Remainder of Rank-1 Matrices..................... 93 6.2.2 A Note on Latin Squares........................ 96 6.2.3 Rank Theorem............................. 98 6.3 p-Adic Matrices................................. 100 6.3.1 Binary Code Matrices......................... 100 6.3.2 Non-Symmetric Matrices........................ 104 6.3.3 Odd Primes............................... 105 7 Conclusion and Future Work 107 References 111 ix Chapter 1 Introduction In this thesis we will study the problem of computing the Smith normal form over local rings, and some related problems regarding computing nullspace vectors, rank properties and eigenvalues properties. We are primarily concerned with the case of sparse matrices, but we will occasionally consider dense matrices as well. We start this chapter by defining several concepts central to our discussion. We finish by giving an overview of the chapters of this thesis. 1.1 Smith Normal Form Let F be a field, and let A be an n × n matrix over F. We use det(A), rank(A), charpoly(A), im(A), ker(A) to denote the determinant, rank, characteristic polynomial, image, and right kernel of A, respectively. We use minpoly(A) to denote the minimal polynomial of A, that is, the lowest degree non-zero monic polynomial f 2 F[x] such that f(A) = 0. We use fλ1; : : : ; λng to denote the eigenvalues of A. Two notions of matrix transformation are notable: similarity transformation and equiv- alence transformation. We say that A and B are similar if there exists an invertible matrix W such that A = W −1BW . On the other hand, we say that A and B are equivalent if there exists two invertible matrices P , Q such that A = P BQ. When working over a principal ideal ring R, we require that P , Q be unimodular, i.e., have a determinant which is unit 1 in the ring. Some matrix invariants, such as rank (when suitably defined over the ring), are preserved under both transformations. Other invariants, such as the characteristic polynomial (and hence) the eigenvalues, are preserved under the similarity transformation only. This will have important implications later when we discuss the preconditioning operations. Consider a matrix A over a field. The eigenvalues of A will typically lie in an ex- tension field K (that is, the splitting field of the characteristic polynomial of A). It is well-known that over K, the matrix A can be brought to the Jordan form using a similarity transformation J = W AW −1. The matrix J is a block-diagonal matrix given by 2 3 J1 6 7 6 7 6 J2 7 J = 6 7 ; 6 .. 7 6 . 7 4 5 J` where each Ji is called the Jordan block, which is a ki × ki matrix 2 3 λi 1 6 7 6 .. .. 7 6 . 7 Ji = 6 7 : 6 .. 7 6 . 1 7 4 5 λi The number of Jordan blocks associated with an eigenvalue and the dimension of each block is determined by the geometric and algebraic multiplicities of that eigenvalue. We note that minpoly(A) is always a factor of charpoly(A). However, as we will discuss later, if all the Jordan blocks corresponding to the zero eigenvalue have size at most 1, then charpoly(A) and minpoly(A) differ by at least a power of x, which will be useful in computing