Topic discussion on Exchangeable Arrays

$ \ BF proposition: $ set $ \ Sigma \ In L \ left ({v, N, C} \ right) $, $ {F _ \ Sigma} \ left (\ Lambda \ right) $ is the feature polynomial of $ \ Sigma $, and $ \ left ({F _ \ Sigma} \ left (\ Lambda \ right), {F'} _ \ Sigma} \ left (\ Lambda \ right )} \ right) = 1 $, then

(1) $ \ Sigma \ Tau = \ Tau \ Sigma $ when and only when $ \ Sigma $ is a feature vector of $ \ Tau $

(2) $ \ Sigma \ Tau = \ Tau \ Sigma $ when and only when $ \ Tau $ is $ {\ Sigma ^ 0}, {\ Sigma ^ 1 }, linear Combination of {\ Sigma ^ 2}, \ cdots, {\ Sigma ^ {n-1} $

(3) $ \ Sigma \ Tau = \ Tau \ Sigma $ A Polynomial $ f (x) $, make $ \ Tau = f \ left (\ Sigma \ right) $

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$ \ BF proposition: $ set $ V $ to $ N $ dimension complex linear space. $ M $ is a non-empty set composed of some linear transformations on $ V $, it is known that the elements in $ M $ do not have non-trivial public invariant subspaces, and the linear transformation $ \ mathcal {B} $ satisfies \ [\ mathcal {A} \ mathcal {B }=\ mathcal {B} \ mathcal {}, \ forall \ mathcal {A} \ in M \] proof: there must be a plural number $ \ Lambda $, making $ \ mathcal {B }=\ Lambda \ mathcal {I} $, $ \ mathcal {I} $ indicates constant transformation.

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$ \ BF proposition: $ set $ {f ^ n} $ to the $ N $ dimension vector space on the number field $ F $ and $ \ Sigma: {f ^ n} \ to {f ^ n} $ is a linear transformation. For any $ A \ in {M_n} \ left (f \ right) $, \ [\ Sigma \ left ({A \ Alpha} \ right) = A \ Sigma \ left (\ Alpha \ right ), \ forall \ Alpha \ in {f ^ n} \]
Proof: $ \ Lambda \ In F $ exists, so that $ \ Sigma = \ Lambda \ cdot ID {f ^ n }$, where $ id {f ^ n} $ is a constant transformation.

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$ \ BF proposition: $ set $ a, B \ in {M_n} \ left (f \ right) $. If $ AB = BA $, when $ B $ is a zero-power array, $ \ left | {a + B} \ right | = \ left | A \ right | $

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$ \ BF proposition: $ set $ \ Sigma, \ Tau \ In L \ left ({v, N, f} \ right) $, and $ {\ Sigma ^ 2 }=\ Sigma $, then $ \ Sigma \ Tau = \ Tau \ Sigma $ is the constant sub-space of $ \ Tau $ and $ im \ Sigma $ only when $ Ker \ Sigma $ and $ im \ Sigma $

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$ \ BF proposition: $ set $ A \ in {R ^ {n \ times N }}$, known $ A $ in $ {R ^ {n \ times N }}$ center sub \ [C \ left (A \ right) =\ left \ {x \ in {R ^ {n \ times N }}| AX = XA} \ right \} \] is $ {R ^ {n \ times N} $'s sub-space, proof: When $ A $ is a real symmetric array, $ \ dim C \ left (A \ right) \ geqslant N $, and the equal sign is valid only when $ A $ has $ N $ different feature values.

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$ \ BF proposition: $ set $ \ Sigma, \ Tau $ to $ N $ Dimension Linear Space $ V $ linear transformation, and each has a base composed of feature vectors, the necessary and sufficient conditions for $ \ Sigma \ Tau = \ Tau \ Sigma $ are the presence of a group of bases in $ V $, make each base vector a common feature vector of $ \ Sigma $ and $ \ Tau $

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$ \ BF proposition: $ if the feature polynomial of $ {A _ {n \ times N }}$ is the same as the least polynomial, $ B $, make $ AB = BA $ if and only if there are times $ \ leqslant n-1 $ polynomials $ f (x) $, make $ B = f (a) $

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Appendix 

$ \ BF proposition: $ set $ a, B \ in {M_n} \ left (f \ right) $, and the matrix $ A $ has different feature values, if $ AB = BA $, $ A, B $ can be similar to the right corner at the same time.

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$ \ BF proposition: $ set $ a, B \ in {M_n} \ left (f \ right) $, and $ A, B $ can be subject to cardification, if $ AB = BA $, $ A, B $ can be similar to the right corner at the same time.

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$ \ BF proposition: $ set $ A, B $ to $ N $ level real symmetric arrays. If $ AB = BA $, orthogonal arrays exist $ q $, so that $ {q ^ {-1} AQ, {q ^ {-1} BQ $ can be similar to Keratin at the same time

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$ \ BF proposition: $ set $ a, B \ in {M_n} \ left (f \ right) $. If $ AB = BA $, a reversible array exists $ p $, make $ {P ^ {-1} AP, {P ^ {-1} BP $ up and down simultaneously

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$ \ BF proposition: $

 

 

 

Topic discussion on Exchangeable Arrays