In this paper, the concepts of con-numbers and con-matrices are proposed by the introduction of conjugate operators into the field of complex numbers, and some properties of these two concepts are derived. In addition, two potential applications of these two concepts are addressed in details. Specifically, a class of quantum systems is expressed in terms of the proposed con-numbers, and the ℝ-linear mapping is represented by con-matrices. These two applications imply the importance of the con-numbers and con-matrices. In addition, some methods are discussed on the construction of the conjugate-operator-induced numbers. Also, some further research directions are provided on these extended numbers and matrices with their applications to systems control.

本文通过将共轭算子引入复数领域来提出对数和矩阵的概念，并推导了这两个概念的一些性质。此外，还将详细介绍这两个概念的两个潜在应用。具体来说，一类量子系统用提出的常数表示，而expressed-线性映射则用矩阵表示。这两个应用暗示了数字和矩阵的重要性。此外，还讨论了一些有关构造共轭算子诱导数的方法。而且，在这些扩展数和矩阵及其在系统控制中的应用方面，提供了一些进一步的研究方向。