Journal of Mathematical Sciences: Advances and Applications
The infinite upper triangular Pascal matrix is T = [( j )i] for 0 ≤ i, j. It is easy to see that any leading principle square submatrix is triangular with determinant 1, hence invertible. In this paper, we investigate the invertibility of arbitrary square submatrices Tr, c comprised of rows r = [r0, … , rm ] and columns c = c0 , … , cm of T. We show that Tr, c is invertible r ≤ c i.e., ri ≤ ci for i = 0, …, m(), or equivalently, iff all diagonal entries are nonzero. To prove this result, we establish a connection between the invertibility of these submatrices and polynomial interpolation. In particular, we apply the theory of Birkhoff interpolation and Pölya systems.
Kersey, Scott N..
"Invertibility of Submatrices of the Pascal Matrix and Birkhoff Interpolation."
Journal of Mathematical Sciences: Advances and Applications, 41 (1): 45-56.
doi: 10.18642/jmsaa_7100121709 source: http://scientificadvances.co.in/admin/img_data/1092/images/JMSAA7100121709ScottKersey.pdf