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Invariance of Rank and Nullity for Linear Combinations of Sum and Product of Generalized Quadratic Matrices

Shuyuan Liu, Meixiang Chen, Xiaoxia Feng, Zhongpeng Yang, Yanping Xie

Abstract


The class of quadratic matrix determined by the quadratic polynomial includes idempotent matrix, involutory matrix. In 2010, Xie Tao showed that the invariance of rank and nullity for linear combinations of sum and product of two quadratic matrices determined by the same quadratic polynomial. We point out that these conclusions do not hold in general. The definition of generalized quadratic matrix was given by R. W. Farebrother and G. Trenkler
in 2005. In this paper, we discuss the rank for linear combinations of sum and product of generalized quadratic matrices in general case, which are independent of the choice of coefficient. As application, it not only generalizes and improves the corresponding conclusions of idempotent matrix and involutory matrix, but also gives a amended form of the research questions studied by Xie tao.

Keywords


generalized quadratic matrix; linear combination; rank; restricted condition

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