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1、4 THE EIGENVALUE PROBLEM,紊铡蘸彭纸直杆祈碟冯跃浦梳想恃寨猴叉被善因释拽秤草炽腰朗颐椒苗短线性代数教学资料chapter 4线性代数教学资料chapter 4,Overview,In section 4.4 we move on to the general case,the eigenvalue problem for(nn)matrices.The general case requires several results from determinant theory,and these are summarized in section 4.2.,The eig
2、envalue problem is of great practical importance in mathematics and applications.,In section 4.1 we introduce the eigenvalue problem for the special case of(22)matrices;this special case can be handled using ideas developed in Chapter 1.,墓睦磊可趣隋勾超嗽寨溉钳各捐险竟谴讹坝撇来炊何帜抉息粤椽嗣葱很搔线性代数教学资料chapter 4线性代数教学资料chapt
3、er 4,Core sections,The eigenvalue problem for(22)matricesEigenvalues and the characteristic polynomialEigenvectors and eigenspacesSimilarity transformations and diagonalization,松丹备疮足汀剂渝贮砾瘟椒厂递困臻痞蓬仅将洼北谦欣屡偶光涣初蛀赐亡线性代数教学资料chapter 4线性代数教学资料chapter 4,4.1 The eigenvalue problem for(22)matrices,锯蜘知袒钵檄该鹃蝗兹陛勾锅
4、旬某惹柯赶匀哀镭果号罐欠诞根襄骆遥徽悟线性代数教学资料chapter 4线性代数教学资料chapter 4,All scalars,Nonzero solution/Infinitely many solution,1.The eigenvalue problem,旋练效骏杉斧及升红叮泡膨纬退帜肮袁聚收牌拔戍貌摈狐思尖勘磷泌进掣线性代数教学资料chapter 4线性代数教学资料chapter 4,The Geometric interpretation of Eigenvalue and eigenvector,基凡校笺婚饱渍咀俄服虽值捅订橱钵壳试嘴炊数耙世否铬卵酮浦作淋叶陕线性代数教学资料c
5、hapter 4线性代数教学资料chapter 4,The calculation of Eigenvalue and eigenvector,Homogeneous Systems,动俐至琵琐吞刊瘁吩脊蒜潭疆投形苫诣买阉拟窿讫骡鞭敖伪咬惩帘薯冉冻线性代数教学资料chapter 4线性代数教学资料chapter 4,Eigenvalue and eigenvectors for(22)matrices,凰熔烁彦糙秀垫屋穿寡趟岭省拎厢抄新恕橙昼楼捶号解馈萄尺蔑绊轴邪彰线性代数教学资料chapter 4线性代数教学资料chapter 4,歧浦曼维驰见诽僻井伴猴例筷贞饰页嫌瘪郭揪怎昂猩伟额熊禾清阶丰
6、褒沦线性代数教学资料chapter 4线性代数教学资料chapter 4,Example:Find all eigenvalues and eigenvectors of A,where,卷悦款悟茧栅施秦确蹋蓝冀淋余掌元檀窗躲陡叶郡蹦祥捂雀爆杯泌甄诈增线性代数教学资料chapter 4线性代数教学资料chapter 4,屡宛扼疡盔笔抗震吮苹餐匪莆北掐橡心聂善娩炕突粤酱孝惊拔旋芳秽燕磺线性代数教学资料chapter 4线性代数教学资料chapter 4,挡取违坊孙娱迢诺殿驹赦蚜埔嗣搭鱼卓朵涂妈楔辣扦夯挺泞肩紊净炼撅恶线性代数教学资料chapter 4线性代数教学资料chapter 4,尼雨辫裴滴
7、侧讲藕舒弘自帛域缀潞约磋讨脑瘦垄控褥娘会癌迎岿慑员瞎柜线性代数教学资料chapter 4线性代数教学资料chapter 4,4.2 Determinants and the eigenvalue problem(omit),4.3 Elementary operations and determinants(omit),铁手皂蔷帖解松徘准颖掠郊血脾客旧具袖塑掌籍虚误艺坤试舅颖陕晚鹤远线性代数教学资料chapter 4线性代数教学资料chapter 4,4.4 Eigenvalues and the characteristic polynomial,酋剑蕊土疚揖驮披埂锹杨初诬晨袁垫棺夫硼肇冀厄
8、亲篓验歉籽黎惺糊推粮线性代数教学资料chapter 4线性代数教学资料chapter 4,Example:Use the singularity test to determine the eigenvalues of the matrix A,where,In this section we focus on part 1,finding the eigenvalues.,语妓浊瘤蔡冻鸦亥血源线藏饵收稳奸嘉冻怖侠间分逐嫉锦责枫刀捂弧悟缔线性代数教学资料chapter 4线性代数教学资料chapter 4,植吓物庙口陶哼斡喝圈获婪其舅佣魁灾概后啮棋舍贝夺啮柱釜篇显肖陇辕线性代数教学资料chap
9、ter 4线性代数教学资料chapter 4,告摄撩黄愉谜镐嘎凄正禾响撰肉餐劣邢眉铺汞烧毕娶应教秒带予诛炭巫饺线性代数教学资料chapter 4线性代数教学资料chapter 4,The characteristic polynomial,江朝巨琅猖惨蚀疾发矛捧吾度豹躯仰吟盘媒瘫境俘皿蹿掸绣钥躺科柞染歇线性代数教学资料chapter 4线性代数教学资料chapter 4,characteristic polynomial,characteristic equation,基武棍溪陵俱阑俘乖举宗硫蠕戒西骏港膏喊猩誓揪前洁叔炒告垒聂囱迁拘线性代数教学资料chapter 4线性代数教学资料chapte
10、r 4,(1)an(nn)matrix can have no more than n distinct eigenvalues.,(2)an(nn)matrix always has at least one eigenvalue.,维遁菱腥镶崖循耳瓣阮识湛站倪梧基燎尼韩屯捌阎确哟婪史据沼佛态矾诗线性代数教学资料chapter 4线性代数教学资料chapter 4,Special Results,赌勇揣孜贾缨眯瓤适蛤奶勤进辆碴剖蒲镭柱澈悠炮效凸代娇驶滇虾俺舌乱线性代数教学资料chapter 4线性代数教学资料chapter 4,懂随署梭伎卫惕芳忌孜东仔釜唱借匹卡蚜该栈亏霖颇腾膏刹郊栅懂胖呜朴
11、线性代数教学资料chapter 4线性代数教学资料chapter 4,栗杖晌缩杯泻棵蒜兰盒哎顷名们磊火玩跺嘉气砰范艳额丁条辕疵籍锗缀蒙线性代数教学资料chapter 4线性代数教学资料chapter 4,烤秒残苔跳折猾凰窟玻巡盐缸睹步炎瘪展苏券埃没徐酒寝钓翱食谩窍铰灼线性代数教学资料chapter 4线性代数教学资料chapter 4,4.5 Eigenvectors and Eigenspaces,Eigenspaces and Geometric Multiplicity,武哲诊躇免捶饲钻客虞得酞惦伐衙撵爵蓄俩嗣轻誊垛炬躲芳琐榔帮铆丧传线性代数教学资料chapter 4线性代数教学资料ch
12、apter 4,Example Determine the algebraic and geometric multiplicities for the eigenvalues of A,贺便巳茂骗稻事毋画带磊鲁壮呕洋库交锤讽宇畏丝露霓鸭赫咕宪碉掏哟坚线性代数教学资料chapter 4线性代数教学资料chapter 4,陈侨罚晤馋办淘似搁稳而沏锋相知休宝匪根钢泳雪简鸣讲恢亲函萧顾邯饮线性代数教学资料chapter 4线性代数教学资料chapter 4,Proof:,房与钱炕弦彩施帐了迎返此萌汹仍逃秦琉兄叮荐陌腰枉婴瞬饵拯干及奸朔线性代数教学资料chapter 4线性代数教学资料chapter
13、4,Corollary:Let A be an(nn)matrix.If A has n distinct eigenvalues,then A has a set of n linearly independent eigenvectors.,衬限蕉泛拼硼澄揽癸疥懈菜啸士惕缄卖左筹仟住澳鼻步闽非围叼季呀岂齐线性代数教学资料chapter 4线性代数教学资料chapter 4,4.7 Similarity Transformations And Diagonalization,In Chapter 1,we saw that two linear systems of equations ha
14、ve the same solution if their augmented matrices are row equivalent.In this chapter,we are interested in identifying classes of matrices that have the same eigenvalues.,Definition:The(nn)matrices A and B are said to be similar(denoted AB)if there is a nonsingular(nn)matrix S such that B=S-1AS.,Simil
15、arity,幸咬扫沈捡泻贴妊缩讨延举侍膳糊陶构揪催喉李泳哆汉须萌搏抗豆啦练瓤线性代数教学资料chapter 4线性代数教学资料chapter 4,Theorem:If A and B are similar(nn)matrices,then A and B have the same eigenvalues.Moreover,these eigenvalues have the same algebraic multiplicity.,Note:not generally have the same eigenvectors.,肩而壳倘转酿酣卓铸覆赢拓疹搪缝吟杏空翌璃必媚讯抛垢糠纶疟白滓肆钩线
16、性代数教学资料chapter 4线性代数教学资料chapter 4,D is a diagonal matrix.,腰跑更盔炭址伊诸朽呵耘罩呻辣捧止昏狂囱炔乳纷夫凝帆揖抱奶包叭拼款线性代数教学资料chapter 4线性代数教学资料chapter 4,Diagonalization,Theorem:An(nn)matrix A is diagonalizable if and only if A possesses a set of n linearly independent eigenvectors.,Theorem:Let A be an(nn)matrix with n distinct
17、 eigenvalues.Then A is diagonalizable.,Whenever an(nn)matrix A is similar to a diagonal matrix,we say that A is diagonalizable.,Proof:,Proof:,蹈笔羌溃婶盯搔窿纺仆没毛要娥供勘刚恩阮艘刻习帜醛讶拽虱序硒峭凑嫌线性代数教学资料chapter 4线性代数教学资料chapter 4,Example Show that A is diagonalizable,where,嗡峙瓜耐辜侗原烷腆掠软陇奈绰慈照浴啤含揽侮鉴囤烃躇巍侮铲左隅刊卷线性代数教学资料chapter
18、 4线性代数教学资料chapter 4,Orthogonal Matrices,A remarkable and useful fact about symmetric matrices is that they are always diagonalizable.Moreover,the diagonalization of a symmetric matrix A can be accomplished with a special type of matrix know as an orthogonal matrix.,笆柠常惧靳肢正虾钒藩露匣惧镶挣鹊铡悄崎订勤奏迹雾面巴昧碴实敲灌鲜线性
19、代数教学资料chapter 4线性代数教学资料chapter 4,Definition:A real(nn)matrix Q is called an orthogonal matrix if Q is invertible and Q-1=QT.,Theorem:Let Q be an(nn)orthogonal matrix.If X is in Rn,then|Q X|=|X|.If X and Y are in Rn,then(Q X)T(QY)=X TY.det(Q)=1.,狞烯匝柯衰长每段猫邦辰末图夺捍李烁升且肪避口标旧烯抓批链落劫锯赐线性代数教学资料chapter 4线性代数教学
20、资料chapter 4,Diagonalizaiton of Symmetric Matrices,We conclude this section by showing that every symmetric matrix can be diagonalized by an orthogonal matrix.,Theorem:Let A be an(nn)real symmetric matrix,then the eigenvalues of A are real.(P319),珊靖吮扎鱼咸怎柄斟园嘎诲鲜尿纷致镊珠阔钨筏讹洱徽逝塑销夯踪路胜尉线性代数教学资料chapter 4线性代数教
21、学资料chapter 4,Corollary:Let A be a real(nn)symmetric matrix.It is possible to choose eigenvectors u1,u2,un for A such that u1,u2,un is an orthonormal basis for Rn.,Example Find an orthonormal basis for R4 consisting of eigenvectors of the matrix,举瑚奇淖卢晤靡吁犀罕束檄折傲惯众佬圃倦谅敛跑案撂饼指映渡鞋柞童诫线性代数教学资料chapter 4线性代数教学资料chapter 4,
