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TRIANGULAR MATRICES AND COMPLETE HOMOGENEOUS SYMMETRIC POLYNOMIALS
Keywords:
triangular matrix, powers of triangular matrix, series of triangular matrices, complete homogeneous symmetric polynomialsDOI:
https://doi.org/10.17654/0972555525022Abstract
In [1], the authors computed the powers of the real upper triangular matrices,
$$
A_n=\left[\begin{array}{ccccc}
a_1 & a_1 & a_1 & \cdots & a_1 \\
0 & a_2 & a_2 & \cdots & a_2 \\
0 & 0 & a_3 & \cdots & a_3 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & a_n
\end{array}\right]
$$
and demonstrated that the resulting matrices have complete homogeneous symmetric polynomials as entries. Those results are extended to infinite matrices and to infinite series of matrices over integral domains. The inverses of the $A_n$ are computed, as are powers of the inverses. The results also are used to produce new proofs of a famous result about complete homogeneous symmetric polynomials, without the use of generating functions.
Received: March 1, 2025
Accepted: April 9, 2025
References
C. Khetchatturat, U. Leerawat and P. Siricharuanun, Powers of some special upper triangular matrices, Thai J. Math. 22(1) (2024), 111-118.
E. F. Cornelius, Jr., Identities for complete homogeneous symmetric polynomials, JP J. Algebra, Number Theory and Applications 21(1) (2011), 109-116.
E. F. Cornelius, Jr., Endomorphisms and product bases of the Baer-Specker group, Int. J. Math. and Math. Sci., 2009, Article 396475.
A. Arafat and M. El-Mikkawy, Novel identities for elementary and complete symmetric polynomials with diverse applications, AIMS Math. 9(9) (2024), 23489-23511.
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