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arXiv · 2303.12176

A formula for the categorical magnitude in terms of the Moore-Penrose pseudoinverse

Stephanie Chen·Juan Pablo Vigneaux
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Abstract

The magnitude of finite categories is a generalization of the Euler characteristic. It is defined using the coarse incidence algebra of rational-valued functions on the given finite category, and a distinguished element in this algebra: the Dirichlet zeta function. The incidence algebra may be identified with the algebra of $n \times n$ matrices over the rational numbers, where $n$ is the cardinality of the underlying object set. The Moore-Penrose pseudoinverse of a matrix is a generalization of the inverse; it exists and is unique for any given matrix over the complex numbers. In this article, we derive a new method for calculating the magnitude of a finite category, using the pseudoinverse of the matrix that corresponds to the zeta function. The magnitude equals the sum of the entries of this pseudoinverse.

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Stephanie Chen, Juan Pablo Vigneaux. 2023-12-13. A formula for the categorical magnitude in terms of the Moore-Penrose pseudoinverse. https://doi.org/10.36045/j.bbms.230331

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