数学与应用数学毕业论文-关于和与积相等的矩阵对.doc

学号：20105031305本科毕业论文学 院 数学与信息科学学院 专 业 数学与应用数学 年 级 2010级 姓 名 王亚辉 论文题目 关于和与积相等的矩阵对 指导教师 张艳艳 职称 讲师 2014年5月3日 11目 录摘 要1关键词1Abstract1Keywords10 前言11 引理及相关定理22 满足的矩阵对的一些性质63 主要结论及证明7参考文献12 关于和与积相等的矩阵对姓名：王亚辉 学号：20105031305数学与信息科学学院 数学与应用数学专业指导教师：张艳艳 职称: 讲师摘 要：满足的矩阵对之间有着密切联系.本文从矩阵的秩、迹、非奇异性、特征值、对角化等方面，讨论了矩阵对的一些性质，并给出满足这种矩阵对条件下的一些特殊矩阵在迹与秩，行列式计算等方面的性质.关键词：特征值；秩；迹；矩阵对；Hermite阵Matrix having equal sum and productAbstract：If matrix pair satisfies the condition , these two matrices have some connections. In this paper, we discusses some properties of the matrix from the rank , trace, invertibility, eigenvalues, diagonalization, and give the nature of some special matrix which satisfies this matrix in trace and rank, the determinant calculation and so on .Keywords：Eigenvalue; Rank; Trace; Matrix pair; Hermite matrix0 前言矩阵的和与乘积是矩阵的两种基本运算，它们的特征值、秩、正定性等方面的关系问题,在理论上和实际应用中都很有意义,例如 矩阵特征值与奇异值估计在矩阵计算、误差分析中有着重要的应用, 因此对矩阵和与乘积的研究得到了许多学者的关注.对于两个阶矩阵,的乘积,一般主要研究它们的可交换性. 但事实上, 矩阵对 ,它们的和与积相等. 这对矩阵在矩阵的秩、特征值和特征向量、正定性、非奇异性等方面都有一些很密切的联系.通过对此题目的探讨不仅可以加深对矩阵的进一步了解同时也将所学知识与实际结合，更加深刻认识特殊矩阵在实际中的重要应用.文中表示阶单位矩阵，为矩阵的秩，表示矩阵的转置，为阶Hermite 矩阵，为矩阵的迹，表示矩阵的共轭转置，和分别为矩阵和的Kronecker积和Hadamard积.以下用表示集合：，即阶矩阵对符合条件.如矩阵和以及和都是符合条件的矩阵对.1 引理及相关定理定义 设且，若，有，则为正定矩阵.定义 设，若，则称为规范矩阵.引理 若是正定矩阵，则.引理2 若，是非奇异矩阵，则是正定矩阵的充要条件是是正定矩阵.引理 是规范矩阵，若，则是正定矩阵.引理4 相似的矩阵有相同的特征值.引理5 阶矩阵，符合条件的充分必要条件是和互为逆矩阵；且若矩阵对符合条件，则及 证明 因为 .即. 又和互为逆矩阵，所以，故.引理6 若矩阵对符合条件，则存在阶非奇异矩阵和，使得.证明 由引理1显然得证.引理 （Hoffmanwieland定理）设，均为实对称阵，它们的特征值分别为：，则，的特征值之间有如下关系成立：引理 （Neumann 不等式）设，的特征值分别为，则 （1）设，的奇异值分别为，则 （2）引理 设是交换族，那么存在一个酉阵，使得对每个，是上三角的.定理1 设、，是正定对称矩阵，则是正定矩阵的充要条件是.证明 若是正定矩阵，由引理2知，是正定矩阵,由引理1得.反之，若，则由引理4得.因 故 因此，是规范矩阵，由引理3知，是正定矩阵.由引理2知是正定矩阵.定理2 若矩阵对符合条件，则（1）矩阵和的特征值均不为1；若是的特征值，则对应的特征，和有公共的特征向量系；（2）可以对角化的充分必要条件是可以对角化，即，可以同时对角化；（3）若有个不同的特征值，存在一个次数不超过的多项式使得,证明 （1）由引理1，即1既不是的特征值，也不是的特征值.设是矩阵的特征向量，对应的特征值是,则，而,故,，所以也是矩阵的特征向量，对应的特征值为 ；若是矩阵的特征向量，同理可证它也是的特征向量,这说明与有公共的特征向量系.(2) 只证必要性.由相似于对角矩阵,因而存在非奇异矩阵,使,为的特征值,所以令则, 为的特征向量,由(1) 知, 也是矩阵B的特征向量,设,为的特征值, ,则,于是相似于对角矩阵.(3) 有个不同的特征值,故可对角化,由(2) 知也可对角化. 令,取多项式,由于互不相同,根据 Lagrange 插值定理可知,存在一个次数不超过的多项式 ,使得 ,则 ,即有,从而 ,定理1得证.推论 设,为正定的Hermite 阵，且满足条件，则存在酉阵，使得和同时为对角阵.定理3 若,都是数域上满足条件的矩阵，若,的特征值都在中，则存在上非奇异矩阵，使得及都是上三角矩阵，即，可同时上三角化.证明 对矩阵的阶数用数学归纳法.当时，结论显然，假定对阶矩阵结论成立，因为，满足条件，则，且与有公共的特征量，不妨，其中，分别为，的特征值，则存在上的阶非奇异矩阵，使得，其中向量，都是阶矩阵.显然，于是根据归纳假设，存在上的阶非奇异矩阵，使得及同时为上三角矩阵.令，则为上三角阵.同理也是上三角矩阵，定理2得证.推论1 设矩阵对满足条件，若是Hermite矩阵，则也是Hermite 矩阵，且存在阶酉矩阵，使得和为对角矩阵.证明 因为是阶Hermite 矩阵，所以存在阶酉矩阵，使得，由定理1中（2）可知，显然，也是Hermite 矩阵，且可同时对角化，推论1得证.推论2 设,是满足条件的正规矩阵，则，都是正规矩阵，且存在酉矩阵，使得和为对角矩阵.推论3 若矩阵对满足条件，则下列条件等价：(i) 非奇异，（ii）非奇异，（iii）或非奇异，（iv）非奇异.推论4 设，是满足条件的正定（或半正定）矩阵，则，及都是正定（或半正定）矩阵. 两个Hermite 矩阵积与其特征值之间的关系问题有著名的Neumann 不等式，两个实对称矩阵和的特征值关系问题有Hoffmanwielandt 定理，故由引理3，引理4及推论1和4，有推论5 设，是满足条件的正定矩阵，则对任意正整数，有.推论 6 若，是满足条件的Hermite 阵，设为的特征值，则为的特征值，为和的全体特征值.2 满足的矩阵对的一些性质性质1 如果，则有（1） ，均为正整数；（2） ，其中是的多项式，即与的多项式可交换；（3） ，为整数；（4） （矩阵二项式定理），为整数；（5） ，为整数；性质2 （1）若且是可逆的，则可交换；（2）若且是可逆的，则可交换.性质3 （1）若且是正交阵，则可交换；（2）若且是正交阵，则可交换.3 主要结论及证明结论1 是正定矩阵，、都是对称矩阵且满足，则是正定矩阵的充要条件是.证明 因为，由引理5得.从而易得，而为实数，由定理1即得结论. 结论2 矩阵，为满足条件的阶Hermite阵，且，则 .上述等式成立当且仅当存在一个具有标准正交列的矩阵和某个使得. 证 明 设是的非零特征值，由Cauchy-Schwarz不等式得：，即.因为，满足所以，即.所以，可交换.又知，均为Hermite阵，所以为Hermite阵. 下面对等号成立进行证明.充分性：若，且为标准正交列的矩阵，则 .从而 必要性：记，则且，为非零矩阵.为Hermite阵，存在酉阵使得，其中,酉阵由矩阵的特征向量正交化，单位化得到,即，且，（）为的非零特征值.所以又知，所以，将上式展开后重新合并可得： 易得 .记，则，.结论3 矩阵，均为阵，且，则 .证明 因为所以 又知，所以.即有 故 . 结论4 设分别有特征值和 且，则存在指标 的一个排列，使得的特征值是.证明 因为，所以.根据引理9可得，它们可以同时上三角化，即存在酉矩阵，使得和都是上三角矩阵，且分别具有对角元及，又有对角元且以它们为特征值，同时，由于相似于，所以它们必定也是的特征值.由结论4可得推论4.1和推论4.2.推论4.1 且，则的各特征值之和是的各特征值和加上的各特征值的和.证明 由结论4得，的特征值为，故的特征值之和为又知的特征值之和为的特征值之和为 显然推论4.1成立.推论4.1的另一种表述 若且，则.推论4.2 若且，和 分别为的特征值，则是非奇异矩阵.证明 由结论4知的特征值是其中，为的一个重新组合.又知，所以，即 均不为零,所以有个非零特征值，故非奇异.结论5 ，且满足，如果对角阵的主对角线上的元素互不相等，那么也是对角阵.证明 设，其中（），（），因为，故，可得，整理得，又因为时，故时，.参考文献1 杨兴东. 矩阵之和的特征值与奇异值估计J.数学杂志,2004,24(3): 263 - 266.2 席博彦,张晓明. 关于矩阵和与矩阵积的特征值的关系J. 数学研究论,2004,24(4): 689 - 696.3 伍俊良,刘飞. 实对称矩阵和与差的一些特征值与F2范数不等式J. 高等学校计算数学学报,2004,26(4): 365 - 370.4 Sha Hu-yun. Estimation of the Eigenvalues of A B for A 0 ,B 0J. Linear Algebra Appl, 1986,73:147 -150.5 黎罗罗. 可交换厄米特矩阵乘积的特征值J.中山大学学报(自然科学版),1999,38 (2): 6 - 9.6 黄涵,周士藩. 对称矩阵和的秩与惯性指数J .宁夏大学学报(自然科版),1991,12 (4): 1 - 5.7 詹仕林. 矩阵乘积的正定性J .安徽大学学报(自然科学版),2003,27(2): 10 - 12.8 史荣昌,魏丰. 矩阵分析M.第二版.北京：北京理工大学出版社,2005.9 王松桂，吴密霞，贾忠贞. 矩阵不等式M.第二版.北京：科学出版社,2006.10 杨奇. 矩阵分析M.北京：机械工业出版社,2005.请您删除一下内容，O(_)O谢谢！Many people have the same mixed feelings when planning a trip during Golden Week. With heaps of time, the seven-day Chinese National Day holiday could be the best occasion to enjoy a destination. However, it can also be the easiest way to ruin how you feel about a place and you may become more fatigued after the holiday, due to battling the large crowds. During peak season, a dream about a place can turn to nightmare without careful planning, especially if you travel with children and older people. As most Chinese people will take the holiday to visit domestic tourist destinations, crowds and busy traffic are inevitable at most places. Also to be expected are increasing transport and accommodation prices, with the possibility that there will be no rooms available. It is also common that youllwait in the line for one hour to get a ticket, and another two hours at the site, to only see a tiny bit of the place due to the crowds. Last year, 428 million tourists traveled in China over the week-long holiday in October. Traveling during this period is a matter that needs thorough preparation. If you are short on time to plan the upcoming Golden Week it may not be a bad idea to avoid some of the most crowded places for now. There is always a place so fascinating that everyone yearns for. Arxan is a place like this. The beauty of Arxan is everlasting regardless of the changing of four seasons. Bestowed by nature, its spectacular seasonal landscape and mountains are just beyond word. Arxan is a crucial destination for the recommended travelling route, China Inner Mongolia Arxan Hailar Manzhouli. It is also the joint of the four prairies across the Sino-Mongolian border, where people gravitate towards the exotic atmosphere mixed with Chinese, Russian, and Mongolia elements. As a historic site for the Yitian Battle, Arxan still embodies the spirit of Genghis Khan. Walking into Arxan, you will be amazed by a kaleidoscope of gorgeous colors all the year round - the Spring azaleas blooming red in the snow, the Summer sea wavering blue in the breeze, the Autumn leaves painted in yellow covering volcanic traces, and the Winter woods shining white on the vast alpine snowscape. Hinggan League Arxan city is situated in the far eastern area of Inner Mongolia Autonomous Region. Its full name Haren Arxan means hot holy water in the Mongolian language. Arxan is a tourism city in the northern frontier with a blend of large forest, grand prairies, vast snowfield, heaven lake cluster, thermium, as well as volcanic cluster. It is a rare and unique ecotourism base filled with healthy sunshine, clean air and unspoiled green. Nestled close to the countrys largest virgin forest, and known for its spring and ecological environment, Arxan is marveled at by many tourists as the purest land on earth. You cannot miss out the Autumn of Arxan. It is definitely the best with brightly-colored scenery full of emotions. Autumn in the northern part of the country comes earlier than the South. A September rain followed by the footprints of Autumn brings more colors to the once emerald green mountain and blooming grassland. Shutterbugs flock to see for themselves the marvel of splendid colors around the mountains and waters, many of whom have travel a long distance and even camp here only to capture a moment of the nature wonder. The silver birch turns golden, while the larch is still proudly green. You will find yourself drowned in the intoxicating red of the wild fruits as well as the glamour of flowers in full blown. And your heart will be lingering on the woods as its time for the wild fruits to ripe. The picturesque Arxan in Autumn is indeed a fairyland only exists in a dream that satisfies all your fantasies. If itrains heavily on Saturday night, some elderly Chinese will say it is because Zhinu, or the Weaving Maid, is crying on the day she met her husband Niulang, or the Cowherd, on the Milky Way. Most Chinese remember being told this romantic tragedy when they were children on Qixi, or the Seventh Night Festival, which falls on the seventh day of the seventh lunar month, which is usually in early August. This year it falls on Saturday, August 2. Folklore Story As the story goes, once there was a cowherd, Niulang, who lived with his elder brother and sister-in-law. But she disliked and abused him, and the boy was forced to leave home with only an old cow for company. The cow, however, was a former god who had violated imperial rules and was sent to earth in bovine form. One day the cow led Niulang to a lake where fairies took a bath on earth. Among them was Zhinu, the most beautiful fairy and a skilled seamstress. The two fell in love at first sight and were soon married. They had a son and daughter and their happy life was held up as an example for hundreds of years in China. Yet in the eyes of the Jade Emperor, the Supreme Deity in Taoism, marriage between a mortal and fairy was strictly forbidden. He ordered the heaven troop to catch Zhinu back. Niulang grew desperate when he discovered Zhinu had been taken back to heaven. Driven by Niulangs misery, the cow told him to turn its hide into a pair of shoes after it died. The magic shoes whisked Niulang, who carried his two children in baskets strung from a shoulder pole, off on a chase after the empress. The pursuit enraged the empress, who took her hairpin and slashed it across the sky creating the Milky Way which separated husband from wife. But all was not lost as magpies, moved by their love and devotion, formed a bridge across the Milky Way to reunite the family. Even the Jade Emperor was touched, and allowed Niulang and Zhinu to meet once a year on the seventh night of the seventh month. This is how Qixi came to be. The festival can be traced back to the Han Dynasty (206 BC-AD 220). Traditionally, people would look up at the sky and find a bright star in the constellation Aquila as well as the star Vega, which are identified as Niulang and Zhinu. The two stars shine on opposite sides of the Milky Way. Customs In bygone days, Qixi was not only a special day for lovers, but also for girls. It is also known as the Begging for Skills Festival or Daughters Festival. In this day, girls will throw a sewing needle into a bowl full of water on the night of Qixi as a test of embroidery skills. If the needle floats on top of the water instead of sinking, it proves the girl is a skilled embr