Let the 
matrix 
satisfy the conditions of the Perron-Frobenius
theorem with spectral radius 
, and let the 
matrix 
satisfy
 |
for 
,
2, ..., 
.
Then any eigenvalue 
of 
satisfies the inequality 
with the equality sign holding only when there
exists an 
diagonal matrix 
(where 
and 
is the Kronecker delta)
such that
 |
