2. (oldenburgere) set $ A \ In M_n $, $ \ rock (a) $ to indicate the spectral radius of $ A $, that is, the creator of the modulus of the feature value of $ A $. proof: $ \ Bex \ vlm {k} a ^ K = 0 \ LRA \ rock (a) <1. \ EEx $
Proof: $ \ rA $: Based on Jordan standard theory, there is a reversible array $ p $, make $ \ Bex P ^ {-1} AP =\sex {\ BA {CCC} \ lm_1 & * \\\\ ddots \\\& \ lm_n \ EA }, \ EEx $ and $ \ Bex P ^ {-1} a ^ Kp = \ sex {\ BA {CCC} \ lm_1 ^ K & * \ ddots & \\& & \ lm_n ^ k \ EA }. \ EEx $ thus, $ \ Bex \ vlm {k} a ^ K = 0 \ Ra \ vlm {k} \ lm_ I ^ K = 0 \ rA | \ lm_ I | <1 \ Ra \ rock (a) <1. \ EEx $ \ la $: $ \ rock (a) <1 $, $ | \ lm_ I | <1 $, $ \ beex \ Bea \ sex {\ BA {CCCC} \ lm & 1 & \\\& \ ddots & \\& \ ddots & 1 \\&&& \ lm \ EA }_{ s \ times s} & = ( \ Lm I-U) ^ k \ quad \ sex {u =\sex {\ BA {CCCC} 0 & 1 & \\\& \ ddots & \\& \ ddots & 1 \\& & 0 \ EA }}\\\& = \ sum _ {I = 0} ^ K c_k ^ I \ lm ^ I (-U) ^ {k-I }\\\& =\ sum _ {I = k-s} ^ kc_k ^ I \ lm ^ I (-U) ^ {k-I} \ Quad (k \ geq s) \ & \ to 0 \ quad \ sex {k \ To \ infty }, \ EEA \ eeex $ the last step is because $ \ Bex \ sev {\ frac {K !} {I! (K-I )!} \ Lm ^ I} \ Leq \ frac {K !} {(K-S )!} | \ Lm | ^ {k-s} <k ^ s | \ lm | ^ {k-s} \ to 0 \ quad \ sex {k \ To \ infty }. \ EEx $
[Zhan Xiang matrix theory exercise reference] exercise 1.2