Topic discussion on rank 1 and full 1 Arrays

Rank $1 $ Matrix

$ \ 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 exercise: $ \ BF (09 Nankai Wu) $ set $ V $ to a number field $ p $ N $ dimension linear space, $ \ mathcal {A} $ is a linear transformation on $ V $, and $ r \ left ({\ Cal a} \ right) = 1 $. proof: if $ \ mathcal {A} $ cannot be cardified, it must be a power zero.

 

All $1 $ Matrices

$ \ 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 (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 (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 (11 South China sci-tech 7) $ J $ indicates $ N $ 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 $ 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 (rank 1 matrix)

$ \ 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 '$, evaluate the $ Jordan $ standard form of $ A $

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Topic discussion on rank 1 and full 1 Arrays