Discussion on power zero and rank 1 Arrays

Power zero Array

$ \ BF proposition: $ set $ N $ level matrix $ A $ meet conditions $ {A ^ k} = 0 $ and $ {A ^ {k-1 }}\ Ne 0 $, returns the maximum and minimum values of the number of linear independent solutions of $ AX = 0 $.

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$ \ BF proposition: $ set $ A \ in {M_n} \ left (f \ right) $, and $ {A ^ {n-1 }}\ Ne 0, {A ^ n} = 0 $, $ x \ in {M_n} \ left (f \ right) $ does not exist, making $ {x ^ 2} = A $

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$ \ BF proposition: $ set $ A ={\ left ({A _ {IJ }}\ right) _ {n \ times N }}$ to a power zero array, and $ {A _ {12 }}\ Ne 0, {A _ {13 }}= 0, {A _ {22 }}= 0, {A _ {23 }}\ Ne 0 $, proof: The Matrix $ B $ does not exist, making $ {B ^ {n-1 }}= a $

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$ \ BF proposition: $

$ \ BF exercise: $ \ BF (10 Nankai JIU) $ set $ V $ to $ N $ dimension complex linear space, $ {\ Text {end }}\ left (V \ right) $ is the linear space composed of all linear transformations on $ V $, and $, B $ is a subset of $ V $, $ A \ subset B $, order \ [M = \ left \ {x \ in {\ Text {end} \ left (V \ right) | xy-Yx \ In, \ forall Y \ In B} \ right \} \] If $ {x_0} \ in M $ meets $ tr \ left ({x_0} y} \ right) = 0, \ forall Y \ in M $, proof: $ x_0 $ must be a power-zero linear transformation

Rank 1 Array

$ \ BF proposition: $ set the real matrix $ A ={\ left ({A_1}, \ cdots, {a_n }}\ right) ^ t} \ left ({A_1}, \ cdots, {a_n }}\ right) $, and $ \ sum \ limits _ {k = 1} ^ n {A_k} ^ 2} = 1 $, proof: $ \ left | {e-2a} \ right | =-1 $

$ \ BF proposition: $ set $ V $ to the $ N $ dimension linear space on the number field $ p $, $ \ mathcal {A} $ to the linear transformation on $ V $, and $ r \ left ({\ Cal a} \ right) = 1 $. Proof: If $ \ mathcal {A} $ cannot be cardified, it must be a power of zero.

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$ \ BF proposition: $ set $ A, B $ to $ N $ square matrix, and $ R (AB-BA) = 1 $, then $ A, B $ can be triangle at the same time

$ \ BF proposition: $ proof $ x = XJ + JX $ only has zero solution, where $ X and J $ are square arrays of $ N $, and all elements of $ J $ are $1 $

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$ \ BF exercise: $ \ BF (12 huake five) $ set all elements of $ A $ to $1 $. Calculate the feature polynomials and least polynomials of $ A $, it is also proved that a reversible array $ p $ exists, so that $ {P ^ {-1} AP $ is a diagonal array.

$ \ BF exercise: $ \ BF (10 huake 6) $ in $ {R ^ n} $ space, known linear transformations $ T $ the coordinates under any base $ {e_ I} $ are $ {\ left ({, \ cdots, 1} \ right) ^ \ Prime} $, $ {e_ I} $ is the column vector of the $ I $ column of the unit array. Evaluate the feature value of $ T $, it is also proved that there is a set of standard orthogonal bases in $ {R ^ n} $, so that the matrix of $ T $ under this base is a diagonal matrix.

$ \ BF exercise: $ \ BF (11 South China sci-tech 7) $ J $ indicates $ N $ level matrix of $1 $ \ left ({n \ Ge 2} \ right) $, set $ f \ left (x \ right) = a + bx $ to the polynomial of the rational number field $ q $, so that $ A = f (j) $

$ (1) $ evaluate all feature values and feature vectors of $ J $

$ (2) $ evaluate all feature subspaces of $ A $

$ (3) $ A $ is it right? If the right corner can be obtained, find a reversible array on $ q $, make $ {P ^ {-1} AP $ a diagonal array, and write this diagonal array.

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Appendix 1 (power zero array) 

$ \ BF Proposition 1: $ A $ is a power zero array. Only when the feature values of $ A $ are all $0 $

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$ \ BF Proposition 2: $ set $ A $ to $ N $ level power zero array, then $ \ left | {e + A} \ right | = 1 $

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$ \ BF Proposition 3: $ set $ A \ in {M_n} \ left (f \ right) $, and for any $ k \ in {z ^ +} $, $ tr \ left ({A ^ k} \ right) = 0 $, then $ A $ is a zero-power array.

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$ \ BF proposition 4: $

Appendix 2 (rank 1)

$ \ BF Proposition 1: $ N $ level matrix $ A $ the rank of $1 $ must be a non-zero column vector $ \ Alpha, \ beta $, make $ A = \ Alpha \ beta '$

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$ \ BF Proposition 2: $ set $ \ Alpha, \ beta $ to $ N $ dimension non-zero column vector, and $ A = \ Alpha \ beta '$, then $ {A ^ 2} = tr \ left (A \ right) \ cdot a $, and then $ {A ^ k} = tr {\ left (A \ right) ^ {k-1 }}\ cdot a $

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$ \ BF Proposition 3: $ set $ \ Alpha, \ beta $ to $ N $ dimension non-zero column vector, and $ A = \ Alpha \ beta '$, evaluate the feature values and feature vectors of $ A $

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$ \ BF proposition 4: $ set $ \ Alpha, \ beta $ to $ N $ dimension non-zero column vector, and $ A = \ Alpha \ beta '$, returns the smallest polynomial of $ A $.

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$ \ BF proposition 5: $ set $ \ Alpha, \ beta $ to $ N $ dimension non-zero column vector, and $ A = \ Alpha \ beta '$, then $ A $ is similar to the diagonal matrix. The required and sufficient conditions are $ tr \ left (A \ right) \ Ne 0 $

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$ \ BF Proposition 6: $ set $ \ Alpha, \ beta $ to $ N $ dimension non-zero column vector, and $ A = \ Alpha \ beta '$, if $ tr () = 0 $, then $ A $ is a power zero Array

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$ \ BF proposition 7: $ set $ \ Alpha, \ beta $ to $ N $ dimension non-zero column vector, and $ A = \ Alpha \ beta '$, evaluate the $ Jordan $ standard form of $ A $

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Discussion on power zero and rank 1 Arrays