Miscellaneous equalities for idempotent matrices with applications

This article brings together miscellaneous formulas and facts on matrix expressions that are composed by idempotent matrices in one place with cogent introduction and references for further study. The author will present the basic mathematical ideas and methodologies of the matrix analytic theory in a readable, up-to-date, and comprehensive manner, including constructions of various algebraic matrix identities composed by the conventional operations of idempotent matrices, and uses of the block matrix method in the derivation of closed-form formulas for calculating the ranks of matrix expressions that are composed by idempotent matrices. The author also determines the maximum and minimum ranks of some matrix pencils composed by the products of matrices and their generalized inverses and uses the ranks to characterize algebraic performance of the matrix pencils.

Witnessing matrix identities and proof complexity

We use results from the theory of algebras with polynomial identities (PI-algebras) to study the witness complexity of matrix identities. A matrix identity of [Formula: see text] matrices over a field [Formula: see text]is a non-commutative polynomial (f(x1, …, xn)) over [Formula: see text], such that [Formula: see text] vanishes on every [Formula: see text] matrix assignment to its variables. For every field [Formula: see text]of characteristic 0, every [Formula: see text] and every finite basis of [Formula: see text] matrix identities over [Formula: see text], we show there exists a family of matrix identities [Formula: see text], such that each [Formula: see text] has [Formula: see text] variables and requires at least [Formula: see text] many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI-algebras together with a generalization of the arguments in [P. Hrubeš, How much commutativity is needed to prove polynomial identities? Electronic colloquium on computational complexity, ECCC, Report No.: TR11-088, June 2011].We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomial-ring axioms [P. Hrubeš and I. Tzameret, The proof complexity of polynomial identities, in Proc. 24th Annual IEEE Conf. Computational Complexity, CCC 2009, 15–18 July 2009, Paris, France (2009), pp. 41–51; Short proofs for the determinant identities, SIAM J. Comput. 44(2) (2015) 340–383], and their subsystems. We identify a decrease in strength hierarchy of subsystems of PI proofs, in which the [Formula: see text]th level is a sound and complete proof system for proving [Formula: see text] matrix identities (over a given field). For each level [Formula: see text] in the hierarchy, we establish an [Formula: see text] lower bound on the number of proof-steps needed to prove certain identities.Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.

Education Towards Differences: Preventing Gender and Sex Discriminations in Subjects in the Age of Development

The present contribution addresses the issue of preventing and combating gender and sex discrimination in subjects in the age of development and, specifically, children. In our society, characterized by an ideological sexist, genderist and heterocentric matrix, identities that cannot placed within the socially constructed and accepted stereotypes of normality are – often – victims of abuses and they are denied equal treatment. In order to raise awareness of the need to respect personal and social differences, more and more initiatives are being created aimed at promoting a non-discriminatory attitude towards gay, lesbian, bisexual, transgender, non-conforming and queer people. Among such initiatives, the construction and narration of stories for children explaining the concepts inherent in the culture of differences, are a valid tool for fostering a climate and a cultural structure free of gender and sex stigmatization and instead aiming at enhancing “difference”, considering it a positive possibility just like socially-accepted states. The essay, after a brief description of the characteristics of language, attitudes and relationship with reality in subjects in the age of development, examines the methodological and ethical approach to be used with children, so as to allow them to understand the possibility of different forms of identity and otherness. Preventing and countering prejudices arising from beliefs and social models that help in reproducing every day, in civil society, discriminations, harassment, and unequal treatment against many people, represents a mission to create and reproduce an inclusive and democratic society.