講義矩陣計算

Note:Thisassignmenthas8problems.Itisnotmandatory,youcanchoosetoturninitornot.TheintentofthisassignmentistohelpyougetfamiliarwithpartofthecontentcoveredinLecture1.Youmaytakethesequestionsasafter-clasercises.Ifyouhavequestions, theinstructororTAs,andwewillrespondassoonasIProveIfAisidempotent,thenI?AisalsoAisinvolutoryifandonlyif(I?A)(I+A)=IfAisinvolutoryandB=(I+A)/2,thenBis(a)Letx∈???and??????=1.Provethat?????2??????isHermitianandunitary,andalso(b)Showthatiftherealvector??≠0,thentheissymmetricand

??=???

(a)Acirculantmatrixisamatrixwhoseentriesinarowarecyclicallyright-shiftedtoformnextrow.Belowisacirculantmatrixof5×5: ??5= Show證明thatcirculantmatricesare[Hint:Usedefinition.Theentry(i,j)of??????istheinnerproductofrowsiandjofAHankelmatricof??×??isamatrixwhoseentriesareconstantalonganti-diagonals,??5×5AToeplitzmatricof??×??isamatrixwhoseentriesareconstantalongdiagonals,i.e.,??????=where???????isoneof??+???1constants.BelowisaToeplitzmatrixof5×5: ??5×5= AreToeplitzmatricesnormalingeneral?[Hint:tryafew2x2LetAbearealunitaryShowthatdet(A)=LetBbeanotherunitaryrealmatrixandassumethatdet(B)=?det(A).ThenshowA+Bis[Hint:multiplyby????,????andusedet(XY)=det(X)det(Y),det(????)=Giventwovectorsu,v∈whataretheeigenvaluesofthematrixwhatisdet(??????)whatisdet(I+??????)whatisdet(A+??????)whenAisnonsingular?(Hint:youcankeepAinthe6.IfA=??1]isarealsquaren×nmatrix,andifN(??)=R(????).Prove6.IfA= [Hint:provethatAx=0onlyhassolutionx=0,i.e.,N(??)=Forthe

A=[ 0 itsrankGivenamatrix ??= WhatisNowlet’supdate??byadding2totheentry(1,2)of??.Sowecangetanew ??=??+

]= ApplySherman-MorrisonformulatofindtheinverseofBfrom???1.Showtheintermediatestepsinyour矩陣分析，2014a)AI.Ab)AAB=(I+A)/2，Bi(a)設(shè)x∈...且......=1。證明 ob)0，(a)1..2..3..4..5..5..1..2..3..4..5=..4..51..2.提示：使用定義。......的條目(i,j)是A的i行和j行的內(nèi) Hankel1，...115×5Hankel.2.3.4.3.45矩陣分析，2014..×..的矩陣是一個其元素沿對角線恒定的矩陣，即 +...1下面是5×5的Toeplitz矩陣：0..1..2..3..4...1..0..1..2..3=...2...1..0..1.Toeplitz矩陣一般來說是正規(guī)的嗎？[提示：嘗試一些2x2證明det(A)=±1Bdet(B)=.det(A)。然后證明A+Bi提示：乘以