The footprint form of a matrix: Definition, properties, and an application

doi:10.1016/j.laa.2022.06.016

Linear Algebra and its Applications

Volume 651, 15 October 2022, Pages 209-229

https://doi.org/10.1016/j.laa.2022.06.016 Get rights and content

Abstract

We propose an alternative to the well-known row echelon form of a matrix, focused on its “footprint”, that is, on the position of both the leading and trailing nonzero entries of each row. When a matrix is in footprint form, its leading entries are all in different positions (as in the echelon form) but, in addition, its trailing entries are also in different positions. This new form has several interesting properties. In particular, a matrix in footprint form has the smallest footprint among the matrices obtainable from it through elementary row operations, and supports an efficient way to compute the rank of any of its submatrices with a particular shape. This is critical for our application: scoring potential variable orders for the decision diagram encoding all solutions to a set of linear integer constraints, with the goal of finding an order that minimizes the decision diagram size. We then experimentally evaluate the effectiveness of this scoring, on several problem instances small enough to compare all possible orders.

Section snippets

Background

It is well known that, given an $r_{o r i g} \times c$ real matrix $A_{o r i g}$ (assumed to have no zero columns), we can apply a sequence of the three elementary row operations, (1) swapping two rows, (2) multiplying a row by a nonzero value, and (3) adding a multiple of a row to another row, to transform it into a left row echelon form (for brevity, simply echelon form) matrix $A_{e}$ . In such form, the first $r \leq \min {c, r_{o r i g}}$ rows are nonzero, the remaining $r_{o r i g} - r$ rows are zero and the leading (i.e., first nonzero)

The footprint form of a matrix

Instead of just requiring an echelon (ladder) shape formed by the leading entries on the bottom left, it is possible to define a different matrix form by focusing on the positions of both the leading and trailing nonzero entries of each row.

First, we need to define the footprint of a matrix. Recall that, unlike for a set, the elements in a multiset are still not ordered, but can appear multiple times, i.e., ${{a, b, a}} = {{a, a, b}}$ and $| {{a, a, b}} | = 3$ .

Definition 1

Given an $r \times c$ matrix $A^{'} \in A$ , its footprint is the

Properties of the footprint form

We now define the concept of a “minimal” footprint of a matrix. To explain this, we first need to define how to compare footprints.

Definition 4

Define binary relation ⪯ over $F$ such that, given matrices $A^{'}$ and $A^{″}$ in $A$ , their footprints $Φ (A^{'}) = {{a_{1}, . . ., a_{r}}}$ and $Φ (A^{″}) = {{b_{1}, . . ., b_{r}}}$ satisfy $Φ (A^{'}) ⪯ Φ (A^{″}) \Leftrightarrow \exists π \in P_{r} s.t. \forall i \in {1, . . ., r}, a_{i} \subseteq b_{π (i)},$ where $P_{r}$ is the set of permutations of $(1, . . ., r)$ . □

Theorem 1

Relation ⪯ is a partial order over $F$ .

Proof

Given $ϕ = Φ (A) = {{a_{1}, . . ., a_{r}}}$ , $ϕ^{'} = Φ (A^{'}) = {{a_{1}^{'}, . . ., a_{r}^{'}}}$ , and $ϕ^{″} = Φ (A^{″}) = {{a_{1}^{″}, . . ., a_{r}^{″}}} \in F$ , we

Reduced footprint form

It is known that any echelon matrix can be transformed into a reduced echelon form using a sequence of elementary row operations, so that not only is it in echelon form, but also its leading entries $l_{1}, . . ., l_{r}$ all are equal to 1 and all the entries above them are equal to 0. Remarkably, this reduced echelon form is canonical, i.e., given $A_{o r i g}$ only one reduced echelon matrix $A_{re}$ can be obtained from it. An example of reduced echelon matrix $A_{re}$ is shown at the top of Fig. 2, where the grayed

An application

We now describe the application that motivated our investigation into the definition and properties of the footprint form.

Conclusion

We defined the footprint form of a matrix, which, unlike the well-known echelon form, considers the positions of both the leading and trailing entries in each row. We proved that the footprint form provides an immediate way to compute the rank of certain sub-matrices, and that it can be made canonical by imposing additional restrictions.

Our work was motivated by a practical application: finding a proxy for the number of nodes in the multivalued decision diagram (MDD) encoding the set of

Declaration of Competing Interest

We have no Competing Interest.

Acknowledgements

We are grateful to Lorenzo Ciardo and Leslie Hogben for their feedback and help on a preliminary version of this manuscript.

References (10)

H. Anton et al.
Elementary Linear Algebra: Applications Version
(2013)
T. Kam et al.
Multi-valued decision diagrams: theory and applications
Mult. Valued Log.
(1998)
R.E. Bryant
Graph-based algorithms for boolean function manipulation
IEEE Trans. Comput.
(1986)
S. Kimura et al.
A parallel algorithm for constructing binary decision diagrams
B. Bollig et al.
Improving the variable ordering of OBDDs is NP-complete
IEEE Trans. Comput.
(1996)

There are more references available in the full text version of this article.

Cited by (1)

Automatic Generation of Optimization Model using Process Mining and Petri Nets for Optimal Motion Planning of 6-DOF Manipulators
2022, IEEE International Conference on Intelligent Robots and Systems

View full text

The footprint form of a matrix: Definition, properties, and an application

Abstract

Section snippets

Background

The footprint form of a matrix

Properties of the footprint form

Reduced footprint form

An application

Conclusion

Declaration of Competing Interest

Acknowledgements

Elementary Linear Algebra: Applications Version

Multi-valued decision diagrams: theory and applications

Mult. Valued Log.

Graph-based algorithms for boolean function manipulation

IEEE Trans. Comput.

A parallel algorithm for constructing binary decision diagrams

Improving the variable ordering of OBDDs is NP-complete

IEEE Trans. Comput.