Invertibility of Submatrices of Pascal's Matrix and Birkhoff Interpolation
This result provides a complete, simple criterion for invertibility of Pascal submatrices, which is of interest to researchers in matrix theory and combinatorics, but the problem is niche and the solution is a direct characterization.
The paper characterizes the invertibility of arbitrary square submatrices of the infinite upper triangular Pascal matrix, proving that such a submatrix is invertible if and only if its diagonal entries are all nonzero (i.e., the row indices are componentwise less than or equal to the column indices).
The infinite upper triangular Pascal matrix is $T = [\binom{j}{i}]$ for $0\leq i,j$. 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 $T_{r,c}$ comprised of rows $r=[r_0,\ldots,r_m]$ and columns $c=[c_0,\ldots,c_m]$ of $T$. We show that $T_{r,c}$ is invertible iff $r \leq c$ (i.e., $r_i \leq c_i$ for $i=0, \ldots, m$), 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 \polya{} systems.