Special discussion on contract standard form

Contract Standard Form

$ \ BF proposition: $ set $ \ Alpha $, $ \ beta $ to a real $ N $ dimension non-zero column vector, evaluate the positive and negative inertial indexes of $ \ Alpha \ beta '{\ RM {+} \ beta \ Alpha' $

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$ \ BF proposition: $ set $ A =\left ({A _ {IJ }}\ right ), B =\left ({B _ {IJ }}\ right) $ all are $ N $ level positive definite arrays, then $ \ BF {Hadamard Product }$ $ H =\left ({A _ {IJ }}{ B _ {IJ }}\ right) $ is also a positive definite array.

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$ \ BF proposition: $ set $ A $ to $ N $ for the first-order real-symmetric semi-Definite Matrix, then $ A ^ * $ semi-definite

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$ \ BF proposition: $ set $ N $ real quadratic form $ f \ left ({X_1}, {X_2}, \ cdots, {x_n }}\ right) the positive and negative inertial exponent of $ is $ p, q (P \ Ge q) $, and $ q $ dimension sub-space $ W $ makes $ f \ left (x \ right) = 0, \ forall x \ in W $

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$ \ BF proposition: $

$ \ BF exercise: $ \ BF (10 beike dajiu) $ set $ f \ left ({X_1}, {X_2}, \ cdots, {x_n} \ right) $ is a quadratic form with a rank of $ N $, there is a $ \ frac {1} {2} \ left ({n-\ left | S \ right |} \ right) on $ {R ^ n} $) $ dimension subspaces $ {v_1} $,

Make the $ \ left ({X_1}, {X_2}, \ cdots, {x_n }}\ right) \ in {v_1} $, $ f \ left ({X_1}, {X_2}, \ cdots, {x_n }}\ right) = 0 $

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$ \ BF exercise: $ \ BF () $

Orthogonal contract standard form

 

$ \ BF proposition: $ set $ A $ to $ N $ level definite matrix, $ \ Alpha, \ beta $ to $ N $ dimension column vector, then $ {\ left ({\ Alpha ^ t} \ beta} \ right) ^ 2} \ Le \ left ({{\ Alpha ^ t} A \ Alpha} \ right) \ left ({{\ beta ^ t} {A ^ {-1 }}\ beta} \ right) $

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$ \ BF proposition: $ set the maximum feature value of the real symmetric matrix $ A $ to the maximum value of $ X 'ax $, where $ x $ takes the unit vector in $ {R ^ n} $.

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$ \ BF proposition: $ set $ A $, $ B $ to a real symmetric semi-Definite Matrix, then $ tr \ left ({AB} \ right) \ le tr \ left (A \ right) \ cdot tr \ left (B \ right) $

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$ \ BF proposition: $ set $ A, B $ to $ N $ level real symmetric arrays, and $ r \ left ({A + \ Lambda B} \ right) = 1 $ for any number $ \ Lambda $, then $ B = 0 $

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$ \ BF proposition: $ set $ A, B $ as the $ N $ square matrix in the real number field, and $ AB + BA = 0 $. proof: if $ A $ is a semi-definite array, $ AB = BA = 0 $

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$ \ BF proposition: $

$ \ BF exercise: $ \ BF (10 Zhejiang University 5) $ set $ A $ to $ N $ level real-symmetric arrays, then there will be idempotent arrays $ {B _ I} $, make $ A = \ sum \ limits _ {I = 1} ^ s {\ Lambda _ I} {B _ I }}$, where $ I = 1, 2, \ cdots, S $

$ \ BF exercise: $ \ BF (13) $ set $ A $ to $ N $ Level Semi-Definite Matrix, then $ \ left | {A + 2013e} \ right | \ Ge {2013 ^ n} $ is valid only when $ A = 0 $

$ \ BF exercise: $ \ BF (06 Emy of Sciences 7) $ set the real quadratic form $ f \ left (x \ right) = x' ax $, $ A $3 \ times 3 $ real symmetric array, and the formula \ [{A ^ 3}-6 {A ^ 2} + 11A-6e = 0 \] is met for trial calculation $ \ mathop {max} \ limits_a \ mathop {max} \ limits _ {\ left \ | x \ right \ | = 1} f \ left (x \ right) $, where $ {\ left \ | x \ right \ | ^ 2 }={ X_1} ^ 2 + {X_2} ^ 2 + {X_3} ^ 2 $

$ \ BF exercise: $ \ BF (12 China Southern Airlines 8) $ set $ A, B $ to $ N $ level real symmetric arrays, and $ A = {B ^ 3} $ proves the following proposition

(1) equations $ AX = 0 $ same solution with $ BX = 0 $

(2) For any real number $ C \ neq0 $, the matrix $ P = C ^ {2} e _ {n} + CB + B ^ {2} $ is a positive definite array.

(3) $ A $ feature vectors are all $ B $ feature vectors.

$ \ BF exercise: $ \ BF (10) $ set $ A $ to $ N $ level real symmetric reversible arrays, then, the necessary and sufficient conditions for $ A $ Positive Definite are any positive definite arrays $ B $, with $ tr (AB)> 0 $

At the same time, contract keralization

$ \ BF proposition: $ set $ A $ as a positive definite array, $ B $ as a real symmetric array, then $ A $, $ B $ can be subject to contract keralization at the same time

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$ \ BF proposition: $ set $ A $, $ B $ to form a real symmetric semi-Definite Matrix. Then, $ A $, $ B $ can be subject to contract keratin at the same time.

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$ \ BF Proposition 1: $ set $ A and B $ to positive definite arrays, then $ \ left | {a + B} \ right | \ Ge \ left | A \ right | + \ left | B \ right | $

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$ \ BF Proposition 2: $ set $ A and B $ to definite and semi-definite arrays, respectively, then $ \ left | {a + B} \ right | \ Ge \ left | A \ right | $

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$ \ BF Proposition 3: $ set $ A, B $ to be a semi-Definite Matrix, then $ \ left | {a + B} \ right | \ Ge \ left | A \ right | + \ left | B \ right | $

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$ \ BF proposition: $ set $ A $ Real Symmetric positive definite, $ B $ Real Symmetric semi-definite, then $ tr \ left ({B {A ^ {-1 }}\ right) tr \ left (A \ right) \ Ge tr \ left (B \ right) $

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$ \ BF exercise: $ \ BF (10 huake 7) $ set $ A $ as a positive definite matrix, $ B $ as a symmetric matrix, then a constant $ C $, makes $ Ca + B $ a definite Array

$ \ BF exercise: $ \ BF (08 huake III) $ set $ a, B \ in {R ^ {n \ times N }}$, and $ A, B, if a-B $ is positive, $ {B ^ {-1 }}- {A ^ {-1 }}$ is also positive.

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$ \ BF exercise: $ \ BF (08 huake five) $ set $ A and B $ to $ N $ and semi-definite arrays, respectively, then $ \ left | A \ right | + \ left | B \ right | \ Le \ left | {a + B} \ right | $ when and only when $ B = 0 $ time equal sign is set up

$ \ BF exercise: $ \ BF (05 $ \ ln \ det \ left (\ cdot \ right) $ it is a concave function in the symmetric positive definite matrix set, that is, for any symmetric positive definite matrix $ A, B $ and $ \ Lambda \ In \ left [{0, 1} \ right] $, \ [\ ln \ det \ left ({\ Lambda A + \ left ({1-\ Lambda} \ right) B} \ right) \ Le \ Lambda \ ln \ det \ left (A \ right) + \ left ({1-\ Lambda} \ right) \ ln \ det \ left (B \ right) \]

 

 

 

 

 

Special discussion on contract standard form