An observation on the determinant of a Sylvester-Kac type matrix

TL;DR

This paper proves a conjecture about the determinant of a Sylvester-Kac type matrix using a lesser-known result and explores an extension of this determinant property.

Contribution

It introduces a novel proof of a conjecture related to Sylvester-Kac matrices and extends the result to a broader class of matrices.

Findings

Confirmed the conjecture on the determinant of Sylvester-Kac matrices.

Extended the determinant result to a more general matrix class.

Provided a new proof leveraging a lesser-known mathematical result.

Abstract

Based on a less-known result, we prove a recent conjecture concerning the determinant of a certain Sylvester-Kac type matrix and consider an extension of it.

Equations

$$J_{n}(z_{0},z_{1})=\left(\begin{array}[]{cccccc}z_{0}+nz_{1}&1&&&&\\ n&z_{0}+(n-2)z_{1}&2&&&\\ &n-1&z_{0}+(n-4)z_{1}&\ddots&&\\ &&\ddots&\ddots&n-1&\\ &&&2&z_{0}-(n-2)z_{1}&n\\ &&&&1&z_{0}-nz_{1}\end{array}\right)\,$$

$$\prod _{k=0}^n\left(z_0-(n-2k)\sqrt{z_1^2+1}\right).$$

$$\pm(n-2k)\sqrt{z_{1}^{2}+1}\,,\quad\mbox{for $k=0,1,\ldots,\lfloor n/2\rfloor$.}$$

$$\left(\begin{array}[]{ccccc}0&1&&&\\ n&0&2&&\\ &n-1&0&\ddots&\\ &&\ddots&\ddots&n\\ &&&1&0\\ \end{array}\right)\,.$$

$$M_{n}(x,y,u,v)=\left(\begin{array}[]{cccccc}x&u&&&&\\ nv&x+y&2u&&&\\ &(n-1)v&x+2y&\ddots&&\\ &&\ddots&\ddots&n-1&\\ &&&2v&x-(n-1)y&nu\\ &&&&v&x+ny\end{array}\right)\,$$

$$\prod _{k=0}^n\left(x+\frac{ny}{2}+\frac{n-2k}{2}\sqrt{y^2+4uv}\right).$$

$$\prod _{k=0}^{\lfloor n/2\rfloor }\left({\left(x+\frac{ny}{2}\right)}^2-\frac{(n-2k{)}^2}{4}(y^2+4uv)\right).$$

$$M^{\pm}_{n}(a,b,r)=\left(\begin{array}[]{ccccccc}nar&b&&&&&\\ na&((n-1)a\pm b)r&2b&&&&\\ &(n-1)a&((n-2)a\pm 2b)r&3b&&&\\ &&(n-2)a&\ddots&\ddots&&\\ &&&\ddots&\ddots&nb&\\ &&&&a&\pm nbr&\\ \end{array}\right).$$

$$\frac{1}{2}\left(nr(a\pm b)+(n-2k)\sqrt{4ab+r^2(a\mp b{)}^2}\right),$$

Full text

1. The Conjecture

Quite recently, in order to find formulas for the determinants of some Lie algebras, Z. Hu and P.B. Zhang proposed in [10] the following conjecture.

Conjecture 1**.**

The determinant of the matrix $(n+1)\times(n+1)$

[TABLE]

is

[TABLE]

Notice that Conjecture 1 is equivalent to state that the eigenvalues of $J_{n}(0,z_{1})$ are

[TABLE]

The matrix $J_{n}(z_{0},z_{1})$ can be easily identified as an extension of the so-called Sylvester-Kac matrix. In fact, setting $z_{1}=0$ we find the characteristic matrix of the Sylvester-Kac matrix, also known as Clement matrix,

[TABLE]

The characteristic polynomial of this matrix (that is, $\det J_{n}(x,0)$) was first conjecture in [19], by the $19$th century British mathematician James Joseph Sylvester celebrated, among other facets, as the founder of the American Journal of Mathematics, in $1878$.

A fully comprehensive list of results on the different proofs for Sylvester’s conjecture and the eigenpairs of non-trivial extensions of the Sylvester-Kac matrix can be found in [1, 2, 3, 4, 5, 6, 8, 7, 9, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21].

The aim of this short note is to prove Conjecture 1 based on a result by W. Chu in [3]. We also provide a general result containing other particular known determinants.

2. An extension to Sylvester-Kac matrix

In $2010$, cleverly based on two generalized Fibonacci sequences, W. Chu proved the following theorem.

Theorem 2.1** ([3]).**

The determinant of the matrix $(n+1)\times(n+1)$

[TABLE]

is

[TABLE]

Of course, the formula for the determinant in Theorem 2.1 can be rewritten as

[TABLE]

Now setting $x=z_{0}+nz_{1}$, $y=-2z_{1}$, and $u=v=1$, we prove immediately Conjecture 1.

Moreover, in the spirit of [1, 8, 9], using Theorem 2.1, we can also conclude the following theorem.

Theorem 2.2**.**

The eigenvalues of

[TABLE]

are

[TABLE]

for $k=0,1,\ldots,n$….