The footprint form of a matrix: Definition, properties, and an application
Section snippets
Background
It is well known that, given an real matrix (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 . In such form, the first rows are nonzero, the remaining 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., and .
Definition 1 Given an matrix , 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 such that, given matrices and in , their footprints and satisfy where is the set of permutations of . □
Theorem 1 Relation ⪯ is a partial order over . Proof Given , , and , 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 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 only one reduced echelon matrix can be obtained from it. An example of reduced echelon matrix 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.
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