Rank of Submatrices of the Pascal Matrix
Scott Kersey

TL;DR
This paper derives a formula for the rank of submatrices of the Pascal matrix, extending previous invertibility conditions, and applies these results to polynomial approximation problems.
Contribution
It introduces a general formula for the rank of Pascal matrix submatrices and provides bases for their row and column spaces, expanding understanding of their structure.
Findings
Derived a formula for the rank of Pascal matrix submatrices
Provided bases for row and column spaces of these submatrices
Applied results to polynomial approximation problems
Abstract
In a previous paper, we derived necessary and sufficient conditions for the invertibility of square submatrices of the Pascal upper triangular matrix. To do so, we established a connection with the two-point Birkhoff interpolation problem. In this paper, we extend this result by deriving a formula for the rank of submatrices of the Pascal matrix. Our formula works for both square and non-square submatrices. We also provide bases for the row and column spaces of these submatrices. Further, we apply our result to one-point lacunary polynomial approximation.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Matrix Theory and Algorithms · Digital Filter Design and Implementation
Rank of Submatrices of the Pascal Matrix
Scott Kersey
Georgia Southern University, USA
Abstract.
In a previous paper, we derived necessary and sufficient conditions for the invertibility of square submatrices of the Pascal upper triangular matrix. To do so, we established a connection with the two-point Birkhoff interpolation problem. In this paper, we extend this result by deriving a formula for the rank of submatrices of the Pascal matrix. Our formula works for both square and non-square submatrices. We also provide bases for the row and column spaces of these submatrices. Further, we apply our result to one-point lacunary polynomial approximation.
Key words and phrases:
Rank, Pascal matrix, Birkhoff interpolation
1991 Mathematics Subject Classification:
15A15, 41A10
Paper appears in: Journal of Mathematical Sciences: Advances and Applications, 42, 1–12 (2016).
1. Introduction
Pascal’s triangle can be represented by the infinite upper triangular matrix
[TABLE]
with if . Submatrices are of the form
[TABLE]
for some selections and of the rows and columns of , respectively. For example,
[TABLE]
While the rank of this matrix is , it is not obvious to see. The main goal in this paper is to provide a formula for determining the rank of such matrices, and secondly to provide bases for the row and column spaces. Later in the paper, we apply our results to a problem in polynomial approximation.
The results in this paper are a generalization of the main result in [3], stated below as in Theorem 2.1. This theorem provides necessary and sufficient conditions for the invertibility of square submatrices of the Pascal matrix. In the proof of that result, we showed that the invertibility of square submatrices of is equivalent to the unique solvability of a two-point Birkhoff interpolation problem. This Birkhoff interpolation problem has been studied in [1, 2, 6, 7], and generalized to lacunary interpolation (see [4, 5]).
In the present paper, we show how to determine the rank of submatrices of the upper triangular Pascal matrix. Our result applies to both square and non-square submatrices. Our main results are stated in Theorem 2.2 and Theorem 3.1. In Algorithm 3.1, we give an algorithm that demonstrates how to compute linearly independent rows and columns. In the final section of this paper, we apply our results to a problem in lacunary polynomial approximation.
2. Rank of Submatrices of the Pascal Matrix
In [3], the following result was derived for square submatrices of the Pascal triangle.
Theorem 2.1** ([3, Theorem 1.1]).**
Let and be indices to the rows and columns of the square submatrix of the Pascal upper triangular matrix. Then, is invertible iff the following equivalent conditions hold:
- •
* (i.e., for all ).*
- •
There is no zero diagonal entry.
Our goal is to generalize this theorem by finding the rank of arbitrary submatrices, square or non-square. These submatrices are defined by sequences of rows and columns of . In general, . To prove our results, we focus on the condition , and to this end make the following definition.
Definition 2.1*.*
We say is an ordered sub-pair for of length if is a subsequence of , is a subsequence of , and (i.e., for ). If , then . We say is maximal if there is no ordered sup-pair of length greater than .
For example, consider and . Then, is an ordered sub-pair of length . But this sub-pair is not maximal since the ordered sub-pair is of length , which, as it turns out, is maximal. Note that is also a maximal sub-pair, hence these need not be unique.
The sub-pairs provided in Definition 2.1 are used to construct invertible and full-rank submatrices of , as we summarize in Theorem 2.2.
Theorem 2.2**.**
Let and be indices to the rows and columns of the Pascal upper triangular matrix. Suppose is an ordered sup-pair of , with and . Then,
- •
* is a invertible submatrix of .*
- •
* is a submatrix of full row rank.*
- •
* is a submatrix of full column rank.*
If is maximal,
- •
The rank of is .
- •
The columns of span the column space of .
- •
The rows of span the row space of .
Proof.
The dimensions of these submatrices follow from , and . Note that . Since is ordered, . By Theorem 2.1, is invertible. The rank of is at least because it contains the invertible matrix as a submatrix. Since it is of dimension , it necessarily has a full row rank. Likewise, has full column rank.
For the second part, suppose is maximal. Since is a submatrix of ,
[TABLE]
Suppose \text{Rank}\big{(}T_{r,c}\big{)}>p+1. Then, there exists rows and columns of such that \text{Rank}\big{(}T_{{\overline{r}},{\overline{c}}}\big{)}=p+2. By Theorem 2.1, , and so is an ordered sup-pair of length . But this contradicts the assumption that is maximal of order . Therefore, \text{Rank}\big{(}T_{r,c}\big{)}\not>p+1, and so \text{Rank}\big{(}T_{r,c}\big{)}=p+1.
Finally, if is maximal, , and so spans the row space of . Likewise, spans the column space of . ∎
3. Computing the Rank of Submatrices of the Pascal Matrix
In this section we show how to compute a maximal ordered sub-pair of . In Algorithm 3.1, we actually compute the indices and of a maximal pair , such that and .
Algorithm 3.1*.*
Compute indices to a maximal ordered sub-pair of .
- •
Input and .
- •
If , set , go to output.
- •
Otherwise, set .
- •
Set
- •
For , set \beta_{i}:=\max\Big{\{}\min\{k:r_{i}\leq c_{k}\},\ \beta_{i-1}+1\big{)}\Big{\}}.
- •
Set .
- •
Set .
- •
Set .
- •
Output , .
The strategy of our algorithm is to pair each with the first such that . In this first occurrence strategy, . Therefore, .
Theorem 3.1**.**
Let and be indices to the rows and columns of the rectangular submatrix of the Pascal upper triangular matrix. Let be determined by Algorithm 3.1. If , then is a matrix of zeros of rank [math]. Otherwise, let and . Then, is a maximal ordered sub-pair of , and the rank of the submatrices is determined by Theorem 2.2.
Proof.
Suppose . By Algorithm 3.1, this occurs when . In this case, for and , and so the elements of are all zero. Hence, is the zero matrix, of rank [math].
Suppose that and are non-empty. This occurs when . Since and , both and are strictly increasing sequences. Since and are also strictly increasing, it follows that and are both strictly increasing. In the algorithm we choose to be the first occurrence such that and . Therefore, the are incremented by the least amount possible, and so is the largest index for an ordered sub-pair of . Hence, is maximal. ∎
In the next result, we construct index sets of the same rank as and with a minimal number of nonzero entries.
Corollary 3.2**.**
Let be an ordered sub-pair of . Let and be the corresponding index sets such that and . Let be the matrix with , and all other entries zero. Then, . If is maximal, then .
Proof.
By Theorem 2.2, the rank of is . This is the number of elements in and , which equals the number of elements in and . The matrix has zeros for all entries except for entry for . But, since and are strictly increasing, these non-zero entries are in different rows and columns. Hence, the dimension of the row and column spaces of are necessarily equal to the number of non-zero elements, which is the length of or . Therefore, the rank of is , which equals the rank of .
In the case that is maximal, we have by Theorem 2.2 that the rank of equals the rank of , which equals the rank of . ∎
We conclude this section with a constructive example.
Example 3.2*.*
Let and . These are the rows and columns for the following submatrix of the Pascal matrix
[TABLE]
Then, and , and
[TABLE]
since but . Then,
[TABLE]
and so
[TABLE]
Hence, . Then, and . Thus, we compute
[TABLE]
By Theorem 2.2,
[TABLE]
and is invertible. Also by this theorem, the matrix
[TABLE]
has linearly independent rows (full row rank), and the matrix
[TABLE]
has linear independent columns (full column rank). The index matrix defined in Corollary 3.2 is nonzero only at entries . These are , and . Hence,
[TABLE]
which is rank .
4. Application to Polynomial Approximation
In [3], we established a connection between submatrices of the Pascal upper triangular matrix and polynomial interpolation. Let and . Let
[TABLE]
with
[TABLE]
and let
[TABLE]
be the power basis with powers for . Then, the matrix
[TABLE]
is a kind of generalized Vandermonde for the one-point lacunary polynomial interpolation problem. This kind of matrix has been considered in [4, 5], but for square matrices. In [3] we showed that this generalized Vandermonde, for the case that , is related to the Pascal matrix as follows:
Proposition 4.1**.**
Let and be selections of the rows and columns of the Pascal upper triangular matrix . Then, .
Hence, by Theorem 3.1, we have the following:
Corollary 4.2**.**
Let and . Assume . Let be an ordered sub-pair of . Then, has full column rank.
Now we will apply our result to polynomial approximation. Let be an ordered sub-pair of of length . We may assume it is maximal, but it doesn’t need to be. Let
[TABLE]
Assume we are given some data, . Since typically , we cannot expect to interpolate. However, by Corollary 4.2, has full column rank, and so we can easily solve the least squares problem
[TABLE]
The unique solution to this problem is
[TABLE]
where is the pseudo-inverse of .
We conclude with an example.
Example 4.1*.*
Let and . In Example 3.2, it was shown that is a maximal ordered sub-pair of . Therefore, is the degree sequence of our lacunary polynomial, and
[TABLE]
Suppose . Then, the coefficient sequence of the least squares solution is
[TABLE]
Hence, the least squares lacunary polynomial is
[TABLE]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] G. D. Birkhoff. General mean value and remainder theorems with applications to mechanical differentiation and integration. Trans. Am. Math. Soc. 1, 107–136, 1906.
- 2[2] D. Ferguson. The Question of Uniqueness for G. D. Birkhoff Interpolation Problems. J. Approx. Th. 2, 1–28, 1969.
- 3[3] S. Kersey. Invertibility of Submatrices of the Pascal Matrix and Birkhoff Interpolation. J. of Mathematical Sciences: Advances and Applications 41(1), 45–56, 2016.
- 4[4] F. Palacios-Qui n ~ ~ n \tilde{\text{n}} onero, P. Rubió-Díaz, J. Díaz-Barrero, J. Rossell. Order regularity of two-node Birkhoff Interpolation with Lacunary Polynomials. Applied Mathematics Letters 22, 386–389 (2009).
- 5[5] F. Palacios-Qui n ~ ~ n \tilde{\text{n}} onero, P. Rubió-Díaz, J. Díaz-Barrero, J. Rossell. Order of regularity for Birkhoff Interpolation with Lacunary Polynomials. Mathematica Aeterna 1 (3), 129–135 (2011).
- 6[6] G. Pölya. Bemerkungen zur Interpolation und zur Näherungstheorie der Balkenbiegung. Z. Angew. Math. Mech. 11, 4445–449, 1931.
- 7[7] J. M. Whittaker. Interpolatory Function Theory. Cambridge University Press (London) , 1935.
