The generalized spectral radius ( ) of a set complex matrices is ( ) = , where = sup{ ( ): each }. The main object of this paper is to study the following problems. Finiteness Conjecture: For each finite set of n n complex matrices, there is some finite k such that ( ) = . Effective Finiteness Conjecture: For any finite set of n n matrices with rational entries, there is some finite k such that ( ) = . Question: Are the Finiteness Conjecture and the Effective Finiteness Conjecture true? Via the undecidability of the Effective Finiteness Conjecture, we show that the answer to the Finiteness Conjecture is negative.
The generalized spectral radius ( ) of a set complex matrices is ( ) = , where = sup{ ( ): each }. The main object of this paper is to study the following problems. Finiteness Conjecture: For each finite set of n n complex matrices, there is some finite k such that ( ) = . Effective Finiteness Conjecture: For any finite set of n n matrices with rational entries, there is some finite k such that ( ) = . Question: Are the Finiteness Conjecture and the Effective Finiteness Conjecture true? Via the undecidability of the Effective Finiteness Conjecture, we show that the answer to the Finiteness Conjecture is negative.