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Abstract In this paper, we first characterize the structure of symmetries J such that a projection P is J -contractive. Then the minimal and maximal elements of the symmetries J with P ⁎ J P ⩽ J (or J P ⩾ 0) are given. Moreover, some formulas between P (2 I − P − P ⁎) + (P (2 I − P − P ⁎) − ) and P (P + P ⁎) − (P (P + P ⁎) + ) are established. [ABSTRACT FROM AUTHOR]

Abstract This paper presents strong connections between four variants of the zero forcing number and four variants of the Grundy domination number. These connections bridge the domination problem and the minimum rank problem, providing a linear algebra approach to the domination problem. We show that several Grundy domination type parameters are bounded above by minimum rank type parameters. We also give a method to calculate the L -Grundy domination number by the Grundy total domination number, giving some linear algebra bounds for the L -Grundy domination number. [ABSTRACT FROM AUTHOR]