TY - GEN

T1 - Invertibility of Submatrices of Pascal's Matrix and Birkhoff Interpolation

AU - Kersey, Scott N.

N1 - Scott N. Kersey. "Invertibility of Submatrices of Pascal's Matrix and Birkhoff Interpolation" 2013
source:source:http://arxiv.org/abs/1303.6159
Available at: http://works.bepress.com/scott_kersey/3

PY - 2013

Y1 - 2013

N2 - The inﬁnite (upper triangular) Pascal matrix is T = [ ji] for 0 ≤ i, j. It is easy to see that that submatrix T (0 : n, 0 : n) is triangular with determinant 1, hence in particular, it is invertible. But what about other submatrices T (r, x) for selections r = [r0, . . . , rd] and x = [x0, . . . , xd] of the rows and columns of T ? The goal of this paper is provide a necessary and suﬃcient condition for invertibility based on a connection to polynomial interpolation. In particular, we generalize the theory of Birkhoﬀ interpolation and P¨olya systems, and then adapt it to this problem. The result is simple: T (r, x) is invertible iﬀ r ≤ x, or equivalently, iﬀ all diagonal entries are nonzero.

AB - The inﬁnite (upper triangular) Pascal matrix is T = [ ji] for 0 ≤ i, j. It is easy to see that that submatrix T (0 : n, 0 : n) is triangular with determinant 1, hence in particular, it is invertible. But what about other submatrices T (r, x) for selections r = [r0, . . . , rd] and x = [x0, . . . , xd] of the rows and columns of T ? The goal of this paper is provide a necessary and suﬃcient condition for invertibility based on a connection to polynomial interpolation. In particular, we generalize the theory of Birkhoﬀ interpolation and P¨olya systems, and then adapt it to this problem. The result is simple: T (r, x) is invertible iﬀ r ≤ x, or equivalently, iﬀ all diagonal entries are nonzero.

KW - Pascal matrix

M3 - Other

ER -