Page 211 - Matrices theory and applications
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10. Approximation of Eigenvalues
194
(b) Show that if 0 <µ < ρ(M) <η, then there exist constants C, C
such that
k 0
k
k
Cµ ≤ M x ≤ C η ,
∀k ∈ IN.
k
(c) Deduce that log Mx converges in the mean to log ρ(M).
13. Let M ∈ M n (CC) be given. Assume that the Gershgorin disk D l is
disjoint from the other disks D m , m = l. Show that the inverse power
method, applied to M −m ll I n , provides an approximate computation
of the unique eigenvalue of M that belongs to D l .
