Determinants of some special matrices

ABSTRACT

Let $p_1,p_2,\dots ,p_n$ be distinct positive real numbers and m be any integer. Every symmetric polynomial $f(x,y)\in C[x,y]$ induces a symmetric matrix ${\left[f(p_i,p_j)\right]}_{i,j=1}^n.$ We obtain the determinants of such matrices with an aim to find the determinants of $P_m={\left[(p_i+p_j{)}^m\right]}_{i,j=1}^n$ and $B_{2m}={\left[(p_i-p_j{)}^{2m}\right]}_{i,j=1}^n$ for $m\in \mathbb{N}$ (where $\mathbb{N}$ is the set of natural numbers) in terms of the Schur polynomials. We also discuss and compute determinant of the matrix $K_m={\left[\frac{p_i^m+p_j^m}{p_i+p_j}\right]}_{i,j=1}^n$ for any integer m in terms of the Schur and skew-Schur polynomials.

Keywords:

• Loewner matrix
• Vandermonde matrix
• Schur polynomials
• Littlewood-Richardson coefficients
• Cauchy matrix

2010 Mathematics Subject Classifications:

• Primary 15A15
• 05E05

Acknowledgments

The authors are grateful to a referee for valuable comments. Thanks are also due to an editor who helped us in improving the exposition of the final version.

Disclosure statement

No potential conflict of interest was reported by the authors.