proof of determinant lower bound of a strict diagonally dominant matrix
Let’s define, for any
Then, by strict diagonally dominance, one has .Let and , so that the i-th row of matrix is equal to the corresponding row of matrix multiplied by . In this way , one has
Now, let be an eigenvalue of , and the corresponding eigenvector
; let moreover be the index of the maximalcomponent
of , i.e.
Of course, by definition of eigenvector, .Writing the p-th characteristic equation, we have:
so that, being ,
In this way, we found that each eigenvalue of is greater than one inabsolute value; for this reason,
Finally,
so that
whence the thesis.
Remark: Perhaps it could be not immediately evident where thehypothesis of strict diagonally dominance is employed in this proof; infact, inequality (*) and (**) would be, in a general case, not valid; they can be stated only because we can assure, by virtue of strictdiagonally dominance, that the final argument of the absolute value () does remain positive.