Given an
n
×
n
n\times n
matrix with integer entries in the range
[
−
h
,
h
]
[-h,h]
, how close can two of its distinct eigenvalues be?
The best previously known examples (Lu [Minimum eigenvalue separation, ProQuest LLC, Ann Arbor, MI, 1992. Thesis (Ph.D.)–University of California, Berkeley; Wilkinson [The algebraic eigenvalue problem, Monographs on Numerical Analysis, The Clarendon Press, Oxford University Press, New York, 1988]) have a minimum gap of
h
−
O
(
n
)
h^{-O(n)}
. Here we give an explicit construction of matrices with entries in
[
0
,
h
]
[0,h]
with two eigenvalues separated by at most
h
−
n
2
/
16
+
o
(
n
2
)
h^{-n^2/16+o(n^2)}
. Up to a constant in the exponent, this agrees with the known lower bound of
Ω
(
(
2
n
)
−
n
2
h
−
n
2
)
\Omega ((2\sqrt {n})^{-n^2}h^{-n^2})
(Mahler [Michigan Math. J. 11 (1964), pp. 257–262]). Bounds on the minimum gap are relevant to the worst case analysis of algorithms for diagonalization and computing canonical forms of integer matrices (e.g. Dey et al. [Bit complexity of Jordan normal form and polynomial spectral factorization, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2023, pp. Art. No. 42]).
In addition to our explicit construction, we show there are many matrices with a slightly larger gap of roughly
h
−
n
2
/
32
h^{-n^2/32}
. We also construct 0-1 matrices which have two eigenvalues separated by at most
2
−
n
2
/
64
+
o
(
n
2
)
2^{-n^2/64+o(n^2)}
.