讲义矩阵计算

Note:Thisassignmenthas8problems.Itisnotmandatory,youcanchoosetoturninitornot.TheintentofthisassignmentistohelpyougetfamiliarwithpartofthecontentcoveredinLecture1.Youmaytakethesequestionsasafter-clasercises.Ifyouhavequestions, theinstructororTAs,andwewillrespondassoonasIProveIfAisidempotent,thenI−AisalsoAisinvolutoryifandonlyif(I−A)(I+A)=IfAisinvolutoryandB=(I+A)/2,thenBis(a)Letx∈ℂ𝑛and𝑥𝐻𝑥=1.Provethat𝐼𝑛−2𝑥𝑥𝐻isHermitianandunitary,andalso(b)Showthatiftherealvector𝑣≠0,thentheissymmetricand

𝐻=𝐼−

(a)Acirculantmatrixisamatrixwhoseentriesinarowarecyclicallyright-shiftedtoformnextrow.Belowisacirculantmatrixof5×5: 𝐶5= Show证明thatcirculantmatricesare[Hint:Usedefinition.Theentry(i,j)of𝐴𝐴𝐻istheinnerproductofrowsiandjofAHankelmatricof𝑚×𝑛isamatrixwhoseentriesareconstantalonganti-diagonals,𝐶5×5AToeplitzmatricof𝑚×𝑛isamatrixwhoseentriesareconstantalongdiagonals,i.e.,𝑎𝑖𝑗=where𝑟𝑗−𝑖isoneof𝑚+𝑛−1constants.BelowisaToeplitzmatrixof5×5: 𝑇5×5= AreToeplitzmatricesnormalingeneral?[Hint:tryafew2x2LetAbearealunitaryShowthatdet(A)=LetBbeanotherunitaryrealmatrixandassumethatdet(B)=−det(A).ThenshowA+Bis[Hint:multiplyby𝐴𝑇,𝐵𝑇andusedet(XY)=det(X)det(Y),det(𝑋𝑇)=Giventwovectorsu,v∈whataretheeigenvaluesofthematrixwhatisdet(𝑢𝑣𝑇)whatisdet(I+𝑢𝑣𝑇)whatisdet(A+𝑢𝑣𝑇)whenAisnonsingular?(Hint:youcankeepAinthe6.IfA=𝐴1]isarealsquaren×nmatrix,andifN(𝐴)=R(𝐴𝑇).Prove6.IfA= [Hint:provethatAx=0onlyhassolutionx=0,i.e.,N(𝐴)=Forthe

A=[ 0 itsrankGivenamatrix 𝐴= WhatisNowlet’supdate𝐴byadding2totheentry(1,2)of𝐴.Sowecangetanew 𝐵=𝐴+

]= ApplySherman-MorrisonformulatofindtheinverseofBfrom𝐴−1.Showtheintermediatestepsinyour矩阵分析，2014a)AI.Ab)AAB=(I+A)/2，Bi(a)设x∈...且......=1。证明 ob)0，(a)1..2..3..4..5..5..1..2..3..4..5=..4..51..2.提示：使用定义。......的条目(i,j)是A的i行和j行的内 Hankel1，...115×5Hankel.2.3.4.3.45矩阵分析，2014..×..的矩阵是一个其元素沿对角线恒定的矩阵，即 +...1下面是5×5的Toeplitz矩阵：0..1..2..3..4...1..0..1..2..3=...2...1..0..1.Toeplitz矩阵一般来说是正规的吗？[提示：尝试一些2x2证明det(A)=±1Bdet(B)=.det(A)。然后证明A+Bi提示：乘以