An approach to solving A(k) = J-I

作者：Wu, YK^*; Li, Q

来源：Linear Algebra and Its Applications, 2003, 373(SUPPL.): 121-142.

DOI：10.1016/S0024-3795(02)00669-9

• matrix equation
• intersection graph
• partitionable graph
• generalized circulant
• near-k-factor factorization
• row set
• cycle-powers

We conjecture that under the permutation similar equivalence relation there are exactly phi(k) solutions A to the matrix equation A(k) = J(d)( 1)(k) - I-d( l)k, where phi is Euler's totient function, d > 1 is an integer, k > 0 is an odd integer, J is the matrix of all ones, I is the identity matrix, and A is an unknown (0, 1) matrix. We present an approach to verify this conjecture. It establishes a connection between the work of solving the matrix equation A(k) = J - I and the problems of both determining the structure of near-k-factor factorizations of cyclic groups and characterizing cycle-powers. We also collect some results about the latter two problems in order to give more insight into this approach.

• 出版日期2003-11-1
• 单位上海交通大学