Timeline for Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?

6 events

when toggle format	what		by	license	comment
Dec 5, 2015 at 10:58	comment	added	Wolfgang		For sure! :-) .
Dec 5, 2015 at 10:32	comment	added	Stefan Kohl♦		@Wolfgang: There appear to be much less counterexamples if one requires the blocks to have pairwise distinct size.
Dec 5, 2015 at 9:18	comment	added	Wolfgang		@LSB.user255259 There are plenty of solutions. I have looked at matrices with only two different block sizes. If the array is $[a_p, b_q]$ (so e.g. $[2,2,2,8,8]=[2_3, 8_2]$), we obtain a certain disctiminant $[a(p-1)+b(q-1)]^2+4ab(p+q-1)$ which must be a square. The numerous solutions of this diophantine equation come in bunches where the $p$'s and $q$'s form arithmetic progressions, e.g. all $[2_{2k-1}, 8_{k-1}]$ or $[1_{3k-1}, 3_k]$ or $[3_{5k+3}, 5_{3k+2}]$ satisfy it.
Dec 5, 2015 at 5:05	comment	added	L S B. user255259		Thank you for giving the counter example. but first part is ture. i.e, if all the block size are same order then the eigenvalues are integers. for the converse part if matrix has two blocks then their eigenvalues are integer iff product of the two blocks order must be perfect square. similarly how can we interpret for number of blocks greater than 3. @Stefan Kohl
Dec 5, 2015 at 4:57	vote	accept	L S B. user255259
Dec 4, 2015 at 19:00	history	answered	Stefan Kohl♦	CC BY-SA 3.0