A Peculiar Lower Bound to the Spectral Radius |7 March. 2020|As a follow up to the previous post, we discuss a lower bound, given as exercise P7.1.14 in (GvL13).
I have never seen this bound before, especially not with the corresponding proof, so here we go.
Given any , we have the following lower bound: which can be thought of as a sharpening of the bound . In the case of a singular , the bound is trivial , but in the general setting it is less obvious. To prove that this holds we need a few ingredients. First, we need that for any we have . Then, we construct two decompositions of , (1) the Jordan Normal form , and (2) the Singular Value Decomposition . Using the multiplicative property of the determinant, we find that Hence, since it follows that and thus we have the desired result.
(GvL13) G.H. Golub and C.F. Van Loan: ‘‘Matrix Computations’’, 2013 John Hopkins University Press. |