Generating all invertible matrices by row operations
arxiv(2024)
摘要
We show that all invertible n × n matrices over any finite field
𝔽_q can be generated in a Gray code fashion. More specifically,
there exists a listing such that (1) each matrix appears exactly once, and (2)
two consecutive matrices differ by adding or subtracting one row from a
previous or subsequent row, or by multiplying or diving a row by the generator
of the multiplicative group of 𝔽_q. This even holds if the addition
and subtraction of each row is allowed to some specific rows satisfying a
certain mild condition. Moreover, we can prescribe the first and the last
matrix if n≥ 3, or n=2 and q>2. In other words, the corresponding flip
graph on all invertible n × n matrices over 𝔽_q is Hamilton
connected if it is not a cycle.
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