[Zhan Xiang matrix theory exercise reference] exercise 1.8

8. It is proved that any compound matrix is similar to a matrix with all the equal corner elements.

 

 

Proof: (1 ). first, it is proved that the matrix with zero trace is similar to the matrix with zero corner elements. perform mathematical induction for order $ N $. when $ n = 1 $, the conclusion is self-explanatory. assume that the conclusion level $ \ Leq n-1 $ is true, when the level is $ N $, $ \ Bex A = (A _ {IJ }), \ quad \ tr a = A _ {11} + \ cdots + A _ {NN} = 0. \ EEx $ if $ \ bee \ label {limit 8_1_1} \ exists \ I, \ st a _ {II} = 0, \ EEE $ \ Bex e_ I ^ * AE _ I = 0, \ quad e_ I = (\ underbrace {0, \ cdots, 1 }_{ I}, \ cdots, 0) ^ t. \ EEx $ expand $ e_ I $ to a group of bases of $ \ BBC ^ N $ \ Bex e_ I, \ Al_2, \ cdots, \ al_n, \ EEx $ \ Bex a (e_ I, \ Al_2, \ cdots, \ al_n) = (e_ I, \ Al_2, \ cdots, \ al_n) \ sex {\ BA {CC} 0 & * \ * & B \ EA }. \ EEx $ Based on inductive assumptions, $ n-1 $ $ makes $ \ Bex V ^ {-1} bv = \ sex {\ BA {CCC} 0 & * \ & \ ddots & \ * & 0 \ EA }. \ EEx $ order $ \ Bex u = (e_ I, \ Al_2, \ cdots, \ al_n) \ sex {\ BA {CC} 1 & 0 \ 0 & V \ EA}, \ EEx $ then $ U $ is a matrix, and $ \ Bex U ^ {-1} Au = \ sex {\ BA {CC} 0 & * \ * & 0 \ EA }. \ EEx $ if \ eqref {limit 8_1_1} is not valid, then $ \ Bex \ exists \ J \ neq k, \ st a _ {JJ} <0 <A _ {kk }. \ EEx $ note $ \ Bex F (t) = [(1-T) E_j + te_k] ^ ta [(1-T) E_j + te_k], \ EEx $ then $ F (t) $ is a continuous function, $ F (0) = A _ {JJ} <0 <A _ {kk} = F (1) $. by the mediation theorem, $ \ Bex \ exists \ T_0 \ In (0, 1), \ st F (T_0) = 0. \ EEx $ note $ \ Bex \ Sen {(1-T) E_j + te_k} _ 2 ^ 2 = (1-T) ^ 2 + t ^ 2> 0, \ EEx $ and $ (1-T) E_j + te_k $ can be unitized and then expanded to a group of bases of $ \ BBC ^ N $, returns to \ eqref {1_8_1_1.

 

(2 ). repeat the question. generally, $ \ Bex a = ai + C, \ quad A = \ frac {1} {n} \ tr a, \ quad \ tr C = 0. \ EEx $ by (1), there is a matrix $ U $, make $ \ Bex U ^ {-1} Cu = \ sex {\ BA {CCC} 0 & * \ ddots & \ * & 0 \ EA} \ rA U ^ {-1} Au = ai + \ sex {\ BA {CCC} 0 & * \ ddots & \ * & 0 \ EA} = \ sex {\ BA {CCC} A & * \\& \ ddots \\\ * & A \ EA }. \ EEx $

[Zhan Xiang matrix theory exercise reference] exercise 1.8