On the rank of quaternion Hankel matrices

TL;DR

This paper proves that quaternion Hankel matrices have equal left and right ranks and explores their relation to linear recurrence relations, with implications for computational methods.

Contribution

It establishes the equality of left and right ranks for quaternion Hankel matrices and links these ranks to linear recurrence relations with quaternion coefficients.

Findings

Left and right ranks of quaternion Hankel matrices are equal.

Relation between Hankel matrices and linear recurrence relations with quaternion coefficients.

Implications for computational methods using low-rank quaternion Hankel matrices.

Abstract

This paper discusses the left and right ranks of quaternion matrices with Hankel structure. While they are in general different for arbitrary quaternion matrices, we show that the left and right ranks of quaternion Hankel matrices are equal. Moreover, we establish the relation between Hankel matrices and the existence of linear recurrence relations with quaternion coefficients and discuss some practical implications for computational methods relying on low-rank properties of quaternion Hankel matrices.