Topic discussion on Equivalent Standard Form

$ \ BF proposition: $ any square matrix $ A $ can be decomposed into reversible arrays $ B $ product of idempotent arrays $ C $

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$ \ BF proposition: $ any square matrix $ A $ can be decomposed into reversible arrays $ B $ product with symmetric arrays $ C $

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$ \ BF proposition: $ set $ a, B \ in {P ^ {n \ times N }}$, and $ r \ left (A \ right) + r \ left (B \ right) \ Le N $, $ N $ order reversible matrix $ M $ exists, making $ AMB = 0 $

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$ \ BF proposition: $ if $ A $ is $ N $, the square matrix of order $ N $ exists, making $ A = ABA, B = Bab $

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$ \ BF proposition: $ set $ A $ to $ m \ times r$ matrix with a rank of $ r$ $ \ left ({M> r} \ right) $, $ B $ is $ r \ times S $ matrix, there is a reversible array $ p $, so that the $ M-r$ row after $ pa $ is all zero

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$ \ BF proposition: $ set $ t \ In L \ left ({v, N, f} \ right) $, $ s \ In L \ left ({v, n, f} \ right) $ to make $ TST = T $

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$ \ BF proposition: $ set $ A \ in {M _ {M \ times N }}\ left (f \ right ), B \ in {M _ {n \ times M }}\ left (f \ right), m \ ge n, \ Lambda \ Ne 0 $, then

\ [{\ RM {}}\ left | {\ Lambda {e_m}-AB} \ right | ={\ Lambda ^ {M-N }}\ left | {\ Lambda {e_n}-Ba} \ right | \]

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$ \ BF proposition: $ set $ A, B, C $ to $ N $ level matrix, and $ AC = CB $, $ r \ left (C \ right) = r$, then $ A $ and $ B $ have at least $ r$ with the same feature value

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$ \ BF proposition: $ set $ A, B $ to $ N $ level matrix, and $ BA = A $, $ r \ left (A \ right) = r \ left (B \ right) $, then $ {B ^ 2} = B $

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$ \ BF proposition: $ set $ {\ Alpha _ 1}, {\ Alpha _ 2}, \ cdots, {\ Alpha _ n} $ is a base of $ {v_n} \ left (f \ right) $, $ A \ in {M _ {n \ times s }}\ left (f \ right) $, and \ [\ left ({{\ beta _ 1 }, {\ beta _ 2}, \ cdots, {\ beta _ s }}\ right) = \ left ({\ Alpha _ 1}, {\ Alpha _ 2 }, \ cdots, {\ Alpha _ n }}\ right) A \] proof: $ \ dim L \ left ({\ beta _ 1}, {\ beta _ 2 }, \ cdots, {\ beta _ s }}\ right) = r \ left (A \ right) $

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$ \ BF proposition: $ set $ A, B $ to $ N $ level matrix. If $ r \ left ({AB} \ right) = r \ left ({Ba} \ right) $ is true for any $ B $, then $ A = 0 $ or $ A $ reversible

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$ \ BF proposition: $ set $ p \ in {f ^ {r \ times M }}, Q \ in {f ^ {n \ times s }$, for any $ A \ in {f ^ {M \ times N }}$, $ PAQ = 0 $ is available. Proof: $ p = 0 or Q = 0 $

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$ \ BF proposition: $ set $ A \ in {m_m} \ left (f \ right), c \ in {M_n} \ left (f \ right) $, for $ B \ in {M _ {Mn} \ left (f \ right) $, $ r \ left ({\ begin {array} {* {20} {c} A & B \ 0 & C \ end {array} \ right) = r \ left (A \ right) + r \ left (C \ right) $, proof: $ A or C $ reversible

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$ \ BF proposition: $ if $ matrix {A _ {M \ times N }}{ B _ {n \ times P }}{ C _ {P \ times Q }}$ rank all $1 $ matrix $ B $ total $1 $, $ A $ indicates the full column rank and $ C $ indicates the full row rank.

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$ \ BF proposition: $ set $ A \ in {M_n} \ left (f \ right), r \ left (A \ right) = r \ left ({A ^ 2 }}\ right) $, there is a reversible array $ p $, make $ A = p \ left ({\ begin {array} {* {20} {c} D & 0 \ 0 & 0 \ end {array} \ right) {P ^ {-1 }}$

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$(04 Zhejiang University 7) $ set $ v = {P ^ {n \ times N }}$ as the linear space on the number field $ p $, set it to $ A, B, c, D \ in {P ^ {n \ times N }}$, for any $ x \ in {P ^ {n \ times N }}$, order \ [\ Sigma \ left (x \ right) = AXB + cx + XD \]

Proof: $ (1) $ \ Sigma $ is a linear transformation on $ V $ (2) $ when $ c = D = 0 $, $ \ Sigma $ the reversible and mandatory condition is $ \ left | {AB} \ right | \ Ne 0 $

 

 

Topic discussion on Equivalent Standard Form