# Twin-width V: linear minors, modular counting, and matrix multiplication

We continue developing the theory around the twin-width of totally ordered binary structures, initiated in the previous paper of the series. We first introduce the notion of parity and linear minors of a matrix, which consists of iteratively replacing consecutive rows or consecutive columns with a linear combination of them. We show that a matrix class has bounded twin-width if and only if its linear-minor closure does not contain all matrices. We observe that the fixed-parameter tractable algorithm for first-order model checking on structures given with an O(1)-sequence (certificate of bounded twin-width) and the fact that first-order transductions of bounded twin-width classes have bounded twin-width, both established in Twin-width I, extend to first-order logic with modular counting quantifiers. We make explicit a win-win argument obtained as a by-product of Twin-width IV, and somewhat similar to bidimensionality, that we call rank-bidimensionality. Armed with the above-mentioned extension to modular counting, we show that the twin-width of the product of two conformal matrices A, B over a finite field is bounded by a function of the twin-width of A, of B, and of the size of the field. Furthermore, if A and B are n × n matrices of twin-width d over 𝔽_q, we show that AB can be computed in time O_d,q(n^2 log n). We finally present an ad hoc algorithm to efficiently multiply two matrices of bounded twin-width, with a single-exponential dependence in the twin-width bound: If the inputs are given in a compact tree-like form, called twin-decomposition (of width d), then two n × n matrices A, B over 𝔽_2, a twin-decomposition of AB with width 2^d+o(d) can be computed in time 4^d+o(d)n (resp. 4^d+o(d)n^1+ε), and entries queried in doubly-logarithmic (resp. constant) time.

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