矩阵论(徐仲)简明教程习题答案

1习习习习题题题题一一一一1.1.1.1.设λ为的任一特征值,,,,则因λλ22−为AAAA=−AAAA22OOOO的特征值,,,,故022=−λλ....即λ=0=0=0=0或2.2.2.2.2.2.2.2.AAAA～BBBB,,,,CCCC～DDDD时,,,,分别存在可逆矩阵PPPP和QQQQ,,,,使得PPPP1−APAPAPAP====BBBB,,,,QQQQ1−CQCQCQCQ====DDDD....令TTTT====⎟⎟⎠⎞⎜⎜⎝⎛QQQQOOOOOOOOPPPP则TTTT是可逆矩阵,,,,且TTTT1−⎟⎟⎠⎞⎜⎜⎝⎛CCCCOOOOOOOOAAAATTTT====⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−−QQQQOOOOOOOOPPPPCCCCOOOOOOOOAAAAQQQQOOOOOOOOPPPP11====⎟⎟⎠⎞⎜⎜⎝⎛DDDDOOOOOOOOBBBB3.3.3.3.设iiiixxxx是对应于特征值iλ的特征向量,,,,则AAAAiiiixxxx====iλiiiixxxx,,,,用1−AAAA左乘得iiiiiiiiiiiixxxxAAAAxxxx1−λ=....即iiiiiiiiiiiixxxxxxxxAAAA11−−λ=故1−iλ是AAAA的特征值,,,,iiii====1,2,1,2,1,2,1,2,,⋯nnnn....4.4.4.4.(1)(1)(1)(1)可以....AAAAEEEE−λ====)2)(1)(1(−+−λλλ,,,,=PPPP⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−104003214,,,,⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=−2111APAPAPAPPPPP....(2)(2)(2)(2)不可以....(3)(3)(3)(3)⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=110101010PPPP,,,,⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=−1221APAPAPAPPPPP....5.5.5.5.(1)(1)(1)(1)AAAA的特征值是0,0,0,0,1,1,1,1,2.2.2.2.故AAAA====－((((bbbb－aaaa))))2=0.=0.=0.=0.从而bbbb====aaaa....又11111−λ−−−−λ−−−−λ=−λaaaaAAAAIIII====)223(22+−−−aλλλ将λ=1,=1,=1,=1,2222代入上式求得AAAA=0.=0.=0.=0.(2)(2)(2)(2)PPPP====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−101010101....6.6.6.6.AAAAIIII−λ====)1()2(2+−λλ,,,,AAAA有特征值2,2,2,2,2,2,2,2,－1.1.1.1.λ=2=2=2=2所对应的方程组(2(2(2(2IIII－AAAA))))xxxx=0=0=0=0有解向量pppp1====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛041,,,,pppp2====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛401λ====－1111所对应的方程组((((IIII++++AAAA))))xxxx=0=0=0=0有解向量pppp3====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛001令PPPP=(=(=(=(pppp,1pppp,2pppp3)=)=)=)=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛140004111,,,,则PPPP1−====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−4416414030121....于是有AAAA100====PPPP⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛122100100PPPP1−====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−⋅−⋅−⋅−−−12412244023012122431100100100100100100100....27777．(1)(1)(1)(1)AAAAIIII−λ====)1(2+λλ====DDDD3((((λ),),),),λIIII－AAAA有2222阶子式172111−−−−λ====λ－4444λ－4444不是DDDD3((((λ))))的因子,,,,所以DDDD2((((λ)=)=)=)=DDDD1((((λ)=1,)=1,)=1,)=1,AAAA的初等因子为λ－1,1,1,1,2λ....AAAA的JordaJordaJordaJordannnn标准形为JJJJ====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−000100001设AAAA的相似变换矩阵为PPPP=(=(=(=(pppp1,,,,pppp2,,,,pppp3),),),),则由APAPAPAP====PJPJPJPJ得⎪⎩⎪⎨⎧==−=23211ppppApApApApApApApApppppApApApAp0000解出PPPP====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−−241231111;;;;(2)(2)(2)(2)因为),2()1()(23−−=λλλDDDD1)()(12==λλDDDDDDDD，故AAAA～JJJJ====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛200010011设变换矩阵为PPPP=(=(=(=(321,,pppppppppppp),),),),则⎪⎩⎪⎨⎧=+==33212112ppppApApApApppppppppApApApApppppApApApAp⇒PPPP====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−502513803(3)(3)(3)(3)),2()1()(23−+=−=λλλλAAAAIIIIDDDD,1)(2+=λλDDDD1)(1=λDDDD....AAAA的不变因子是,11=dddd,12+=λdddd)2)(1(3−+=λλddddAAAA～JJJJ====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−211因为AAAA可对角化，可分别求出特征值－1111，2222所对应的三个线性无关的特征向量：当λ====－1111时，解方程组,0)(=+xxxxAAAAIIII求得两个线性无关的特征向,1011⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=pppp⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=0122pppp当λ=2=2=2=2时，解方程组,0)2(=−xxxxAAAAIIII得⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=1123pppp,,,,PPPP====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−101110221(4)(4)(4)(4)因⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−+=−41131621λλλλAAAAIIII～⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−2)1(11λλ,,,,故AAAA～JJJJ====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛10111设变换矩阵为PPPP====),,(321pppppppppppp,,,,则⎪⎩⎪⎨⎧+===3232211ppppppppApApApApppppApApApApppppApApApAp21,pppppppp是线性方程组0000=−xxxxAAAAIIII)(的解向量，此方程仴的一般解形为pppp====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+−tsts3取⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=0111pppp,,,,⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=1032pppp为求滿足方程23)(ppppppppAAAAIIII−=−的解向量3pppp,,,,再取,2pppppppp=根据3⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−−−tsts3113113622～⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−tstss00033000311由此可得ssss====tttt,,,,从而向量T3213),,(xxx=pppp的坐标应満足方程sxxx−=−+3213取T3)0,0,1(−=pppp,,,,最后得PPPP====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−0100011318.8.8.8.设ffff((((λ)=)=)=)=4322458−++−λλλλ....AAAA的最小多项式为12)(3+−=λλλAAAAm,,,,作带余除法得ffff((((λ)=()=()=()=(149542235−+−+λλλλ),),),),)(λAm++++1037242+−λλ,,,,于是ffff((((AAAA)=)=)=)=IIIIAAAAAAAA1037242+−====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−346106195026483....9.9.9.9.AAAA的最小多项式为76)(2+−=λλλAm,,,,设ffff((((λ)=)=)=)=372919122234+−+−λλλλ,,,,则ffff((((λ)=)=)=)=)()52(2λλAAAAm+++++2+λ....于是[[[[ffff((((AAAA)])])])]1−====1)2(−+IIIIAAAA....由此求出[[[[ffff((((AAAA)])])])]1−====⎟⎟⎠⎞⎜⎜⎝⎛−321723110.10.10.10.(1)(1)(1)(1)λIIII－AAAA====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−[phone]λλλ标准形⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−2)1(00010001λλ,,,,AAAA的最小多项式为2)1(−λ;;;;2)2)2)2))1)(1(+−λλ;;;;(3)(3)(3)(3)2λ....11.11.11.11.将方程组写成矩阵形式::::⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−=⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛321321188034011ddddddxxxtxtxtx,,,,⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=321xxxxxxxxxxxxxxxx,,,,⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎝⎛=txtxtxtdddddddd321xxxx,,,,AAAA====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−188034011则有JJJJ====PAPPAPPAPPAP1−====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−100010011,,,,....其中PPPP====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛124012001....令x=Pyx=Pyx=Pyx=Py,,,,将原方程组改写成::::,ddJyJyJyJyyyyy=t则⎪⎪⎪⎩⎪⎪⎪⎨⎧−=+==3321211ddddddytyyytyyty解此方程组得::::yyyy1====CCCC1eeeet++++CCCC2TTTTeeeet,,,,yyyy2====CCCC2eeeet,,,,yyyy3====CCCC3eeeet−....于是xxxx====PyPyPyPy====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛++++++−tttttttc)t(cc)t(cctccee24e4e12e2ee3212121....12.12.12.12.(1)(1)(1)(1)AAAA是实对称矩阵....AAAAIIII−λ====2)1)(10(−−λλ,,,,AAAA有特征值10,10,10,10,2,2,2,2,2.2.2.2.当λ=10=10=10=10时....对应的齐次线性方程组(10(10(10(10IIII－AAAA))))xxxx=0=0=0=0的系数矩阵4⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−542452228～⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛000110102由此求出特征向量pppp1=(=(=(=(－1,1,1,1,－2,2,2,2,2)2)2)2)T,,,,单位化后得eeee1====((((32,32,31−−))))T....当λ=1=1=1=1时,,,,对应的齐次线性方程组((((IIII－AAAA))))xxxx=0=0=0=0的系数矩阵⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−−−−−442442221～⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−000000221由此求出特征向量pppp2=(=(=(=(－2,2,2,2,1,1,1,1,0)0)0)0)T,,,,pppp3=(2,=(2,=(2,=(2,0,0,0,0,1)1)1)1)T....单位化后得eeee2=(=(=(=(0,51,52−))))T,,,,eeee3=(=(=(=(535,534,532))))T....令UUUU====⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛−−−53503253451325325231,,,,则UUUU1−AUAUAUAU====⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛1110....(2)(2)(2)(2)AAAA是HermitHermitHermitHermit矩阵....同理可求出相似变换矩阵UUUU====⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛−−−