On ideals of minors of matrices with indeterminate entries

MatrixEponenential-指数矩阵计算

is invertible then .

symmetric, and that if X is skew-symmetric then e X is orthogonal. exp(X*) = (e X)*, where X* denotes the conjugate transpose of X. It follows that if X is Hermitian then e X is also Hermitian, and that if X is skew-Hermitian then e X is unitary. Linear differential equations One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear ordinary differential equations. Indeed, it follows from equation (1) below that the solution of where A is a matrix, is given by The matrix exponential can also be used to solve the inhomogeneous equation See the section on applications below for examples. There is no closed-form solution for differential equations of the form where A is not constant, but the Magnus series gives the solution as an infinite sum. The exponential of sums We know that the exponential function satisfies e x + y = e x e y for any numbers x and y. The same goes for commuting matrices: If the matrices X and Y commute (meaning that XY = YX), then However, if they do not commute, then the above equality does not necessarily hold. In that case, we can use the Baker-Campbell-Hausdorff formula to compute e X + Y. The exponential map Note that the exponential of a matrix is always a non-singular matrix. The inverse of e X is given by e-X. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map from the space of all n×n matrices to the general linear group, i.e. the group of all non-singular matrices. In fact, this map is surjective which means that every non-singular matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field C of complex numbers and not R). The matrix logarithm gives an inverse to this map. For any two matrices X and Y, we have

on the contrary的解析

On the contrary Onthecontrary, I have not yet begun. 正好相反，我还没有开始。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, the instructions have been damaged. 反之，则说明已经损坏。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, I understand all too well. 恰恰相反，我很清楚 https://www.wendangku.net/doc/40725856.html, Onthecontrary, I think this is good. ⑴我反而觉得这是好事。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, I have tons of things to do 正相反，我有一大堆事要做 Provided by jukuu Is likely onthecontrary I in works for you 反倒像是我在为你们工作 https://www.wendangku.net/doc/40725856.html, Onthecontrary, or to buy the first good. 反之还是先买的好。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, it is typically american. 相反，这正是典型的美国风格。 222.35.143.196 Onthecontrary, very exciting.

恰恰相反，非常刺激。 https://www.wendangku.net/doc/40725856.html, But onthecontrary, lazy. 却恰恰相反，懒洋洋的。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, I hate it! 恰恰相反，我不喜欢！ https://www.wendangku.net/doc/40725856.html, Onthecontrary, the club gathers every month. 相反，俱乐部每个月都聚会。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, I'm going to work harder. 我反而将更努力工作。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, his demeanor is easy and nonchalant. 相反，他的举止轻松而无动于衷。 https://www.wendangku.net/doc/40725856.html, Too much nutrition onthecontrary can not be absorbed through skin. 太过营养了反而皮肤吸收不了. https://www.wendangku.net/doc/40725856.html, Onthecontrary, I would wish for it no other way. 正相反，我正希望这样 Provided by jukuu Onthecontrary most likely pathological. 反之很有可能是病理性的。 https://www.wendangku.net/doc/40725856.html, Onthecontrary, it will appear clumsy. 反之，就会显得粗笨。 https://www.wendangku.net/doc/40725856.html,

英语造句

一般过去式 时间状语：yesterday just now (刚刚) the day before three days ag0 a week ago in 1880 last month last year 1. I was in the classroom yesterday. I was not in the classroom yesterday. Were you in the classroom yesterday. 2. They went to see the film the day before. Did they go to see the film the day before. They did go to see the film the day before. 3. The man beat his wife yesterday. The man didn’t beat his wife yesterday. 4. I was a high student three years ago. 5. She became a teacher in 2009. 6. They began to study english a week ago 7. My mother brought a book from Canada last year. 8.My parents build a house to me four years ago . 9.He was husband ago. She was a cooker last mouth. My father was in the Xinjiang half a year ago. 10.My grandfather was a famer six years ago. 11.He burned in 1991

Invasion Assay Using 3D Matrices

Invasion Assay Using 3D Matrices 1. Overview Scientists have developed 3D models to more accurately study cell invasion and migration processes. While most traditional cell culture systems are 2D, cells in our tissues exist within a 3D network of molecules known as the extracellular matrix or ECM. While many of the mechanistic processes required for cell motility in 2D and 3D are similar, factors such as the reduced stiffness of ECM compared to plastic surfaces, the addition of a third dimension for migration, and the physical hindrance of moving through the mesh of long polymers in the ECM, all present different challenges to the cell compared to two-dimensional migration. This video will briefly introduce the basic function and structure of the ECM, as well as the mechanisms by which cells modulate and migrate through it. Next, we’ll discuss a general protocol used to study endothelial cell invasion. Finally, we will highlight several applications of 3D matrices to studying different biological questions. 2. ECM Composition and Cell-ECM Interaction Let’s begin by examining the composition of the ECM, and how cells interact with it.

学生造句--Unit 1

●I wonder if it’s because I have been at school for so long that I’ve grown so crazy about going home. ●It is because she wasn’t well that she fell far behind her classmates this semester. ●I can well remember that there was a time when I took it for granted that friends should do everything for me. ●In order to make a difference to society, they spent almost all of their spare time in raising money for the charity. ●It’s no pleasure eating at school any longer because the food is not so tasty as that at home. ●He happened to be hit by a new idea when he was walking along the riverbank. ●I wonder if I can cope with stressful situations in life independently. ●It is because I take things for granted that I make so many mistakes. ●The treasure is so rare that a growing number of people are looking for it. ●He picks on the weak mn in order that we may pay attention to him. ●It’s no pleasure being disturbed whena I settle down to my work. ●I can well remember that when I was a child, I always made mistakes on purpose for fun. ●It’s no pleasure accompany her hanging out on the street on such a rainy day. ●I can well remember that there was a time when I threw my whole self into study in order to live up to my parents’ expectation and enter my dream university. ●I can well remember that she stuck with me all the time and helped me regain my confidence during my tough time five years ago. ●It is because he makes it a priority to study that he always gets good grades. ●I wonder if we should abandon this idea because there is no point in doing so. ●I wonder if it was because I ate ice-cream that I had an upset student this morning. ●It is because she refused to die that she became incredibly successful. ●She is so considerate that many of us turn to her for comfort. ●I can well remember that once I underestimated the power of words and hurt my friend. ●He works extremely hard in order to live up to his expectations. ●I happened to see a butterfly settle on the beautiful flower. ●It’s no pleasure making fun of others. ●It was the first time in the new semester that I had burned the midnight oil to study. ●It’s no pleasure taking everything into account when you long to have the relaxing life. ●I wonder if it was because he abandoned himself to despair that he was killed in a car accident when he was driving. ●Jack is always picking on younger children in order to show off his power. ●It is because he always burns the midnight oil that he oversleeps sometimes. ●I happened to find some pictures to do with my grandfather when I was going through the drawer. ●It was because I didn’t dare look at the failure face to face that I failed again. ●I tell my friend that failure is not scary in order that she can rebound from failure. ●I throw my whole self to study in order to pass the final exam. ●It was the first time that I had made a speech in public and enjoyed the thunder of applause. ●Alice happened to be on the street when a UFO landed right in front of her. ●It was the first time that I had kept myself open and talked sincerely with my parents. ●It was a beautiful sunny day. The weather was so comfortable that I settled myself into the

英语句子结构和造句

高中英语~词性~句子成分~语法构成 第一章节：英语句子中的词性 1.名词：n. 名词是指事物的名称，在句子中主要作主语.宾语.表语.同位语。 2.形容词;adj. 形容词是指对名词进行修饰~限定~描述~的成份，主要作定语.表语.。形容词在汉语中是（的）.其标志是: ous. Al .ful .ive。. 3.动词：vt. 动词是指主语发出的一个动作，一般用来作谓语。 4.副词：adv. 副词是指表示动作发生的地点. 时间. 条件. 方式. 原因. 目的. 结果.伴随让步. 一般用来修饰动词. 形容词。副词在汉语中是（地）.其标志是：ly。 5.代词：pron. 代词是指用来代替名词的词，名词所能担任的作用，代词也同样.代词主要用来作主语. 宾语. 表语. 同位语。 6.介词：prep.介词是指表示动词和名次关系的词，例如：in on at of about with for to。其特征：

介词后的动词要用—ing形式。介词加代词时，代词要用宾格。例如：give up her（him）这种形式是正确的，而give up she（he）这种形式是错误的。 7.冠词：冠词是指修饰名词，表名词泛指或特指。冠词有a an the 。 8.叹词：叹词表示一种语气。例如：OH. Ya 等 9.连词：连词是指连接两个并列的成分，这两个并列的成分可以是两个词也可以是两个句子。例如：and but or so 。 10.数词：数词是指表示数量关系词，一般分为基数词和序数词 第二章节：英语句子成分 主语：动作的发出者，一般放在动词前或句首。由名词. 代词. 数词. 不定时. 动名词. 或从句充当。 谓语：指主语发出来的动作，只能由动词充当，一般紧跟在主语后面。 宾语：指动作的承受着，一般由代词. 名词. 数词. 不定时. 动名词. 或从句充当. 介词后面的成分也叫介词宾语。 定语：只对名词起限定修饰的成分，一般由形容

六级单词解析造句记忆MNO

M A: Has the case been closed yet? B: No, the magistrate still needs to decide the outcome. magistrate n.地方行政官，地方法官，治安官 A: I am unable to read the small print in the book. B: It seems you need to magnify it. magnify vt.１．放大，扩大；２．夸大，夸张 A: That was a terrible storm. B: Indeed, but it is too early to determine the magnitude of the damage. magnitude n.１．重要性，重大；２．巨大，广大 A: A young fair maiden like you shouldn’t be single. B: That is because I am a young fair independent maiden. maiden n.少女，年轻姑娘，未婚女子 a.首次的，初次的 A: You look majestic sitting on that high chair. B: Yes, I am pretending to be the king! majestic a.雄伟的，壮丽的，庄严的，高贵的 A: Please cook me dinner now. B: Yes, your majesty, I’m at your service. majesty n.１．［Ｍ－］陛下（对帝王，王后的尊称）；２．雄伟，壮丽，庄严 A: Doctor, I traveled to Africa and I think I caught malaria. B: Did you take any medicine as a precaution? malaria n.疟疾 A: I hate you! B: Why are you so full of malice? malice n.恶意，怨恨 A: I’m afraid that the test results have come back and your lump is malignant. B: That means it’s serious, doesn’t it, doctor? malignant a.１．恶性的，致命的；２．恶意的，恶毒的 A: I’m going shopping in the mall this afternoon, want to join me? B: No, thanks, I have plans already. mall n.（由许多商店组成的）购物中心 A: That child looks very unhealthy. B: Yes, he does not have enough to eat. He is suffering from malnutrition.

base on的例句

意见应以事实为根据. 3 来自辞典例句 192. The bombers swooped ( down ) onthe air base. 轰炸机 突袭 空军基地. 来自辞典例句 193. He mounted their engines on a rubber base. 他把他们的发动机装在一个橡胶垫座上. 14 来自辞典例句 194. The column stands on a narrow base. 柱子竖立在狭窄的地基上. 14 来自辞典例句 195. When one stretched it, it looked like grey flakes on the carvas base. 你要是把它摊直, 看上去就象好一些灰色的粉片落在帆布底子上. 18 来自辞典例句 196. Economic growth and human well - being depend on the natural resource base that supports all living systems. 经济增长和人类的福利依赖于支持所有生命系统的自然资源. 12 1 来自辞典例句 197. The base was just a smudge onthe untouched hundred - mile coast of Manila Bay. 那基地只是马尼拉湾一百英里长安然无恙的海岸线上一个硝烟滚滚的污点. 6 来自辞典例句 198. You can't base an operation on the presumption that miracles are going to happen. 你不能把行动计划建筑在可能出现奇迹的假想基础上.

英语造句大全

英语造句大全English sentence 在句子中，更好的记忆单词！ 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character. （2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n.

A metric for covariance matrices

A Metric for Covariance Matrices Wolfgang F¨o rstner and Boudewijn Moonen Diese S¨a tze f¨u hren dahin,die Theorie der krummen Fl¨a chen aus einem neuen Gesichtspunkt zu betrachten,wo sich der Untersuchung ein weites noch ganz unangebautes Feld ¨o ?net ...so begreift man,dass zweierlei wesentlich verschiedene Relationen zu unterscheiden sind,theils nemlich solche,die eine bestimmte Form der Fl¨a che im Raume voraussetzen,theils solche,welche von den verschiedenen Formen ...unabh¨a ngig sind.Die letzteren sind es,wovon hier die Rede ist ...man sieht aber leicht,dass eben dahin die Betrachtung der auf der Fl¨a che construirten Figuren,...,die Verbindung der Punkte durch k¨u rzeste Linien ?]u.dgl.geh¨o rt.Alle solche Untersuchungen m¨u ssen davon ausgehen,dass die Natur der krummen Fl¨a che an sich durch den Ausdruck eines unbestimmten Linearelements in der Form √(Edp 2+2F dpdq +Gdq 2)gegeben ist ... Carl Friedrich Gauss Abstract The paper presents a metric for positive de?nite covariance matrices.It is a natural expression involving traces and joint eigenvalues of the matrices.It is shown to be the distance coming from a canonical invariant Riemannian metric on the space Sym +(n,R )of real symmetric positive de?nite matrices In contrast to known measures,collected e.g.in Grafarend 1972,the metric is invariant under a?ne transformations and inversion.It can be used for evaluating covariance matrices or for optimization of measurement designs. Keywords:Covariance matrices,metric,Lie groups,Riemannian manifolds,exponential map-ping,symmetric spaces 1Background The optimization of geodetic networks is a classical problem that has gained large attention in the 70s. 1972E.W.Grafarend put together the current knowledge of network design,datum transfor-mations and arti?cial covariance matrices using covariance functions in his classical monograph [?];see also [?].One critical part was the development of a suitable measure for comparing two covariance matrices.Grafarend listed a dozen measures.Assuming a completely isotropic network,represented by a unit matrix as covariance matrix,the measures depended on the eigenvalues of the covariance matrix. 1983,11years later,at the Aalborg workshop on ’Survey Control Networks’Schmidt [?]used these measures for ?nding optimal networks.The visualization of the error ellipses for a single point,leading to the same deviation from an ideal covariance structure revealed de?ciencies of these measures,as e.g.the trace of the eigenvalues of the covariance matrix as quality measure ?]emphasized by the authors

(完整版)主谓造句

主语+谓语 1. 理解主谓结构 1) The students arrived. The students arrived at the park. 2) They are listening. They are listening to the music. 3) The disaster happened. 2．体会状语的位置 1) Tom always works hard. 2) Sometimes I go to the park at weekends.. 3) The girl cries very often. 4) We seldom come here. The disaster happened to the poor family. 3. 多个状语的排列次序 1) He works. 2) He works hard. 3) He always works hard. 4) He always works hard in the company. 5) He always works hard in the company recently. 6) He always works hard in the company recently because he wants to get promoted. 4. 写作常用不及物动词 1. ache My head aches. I’m aching all over. 2. agree agree with sb. about sth. agree to do sth. 3. apologize to sb. for sth. 4. appear (at the meeting, on the screen) 5. arrive at / in 6. belong to 7. chat with sb. about sth. 8. come (to …) 9. cry 10. dance 11. depend on /upon 12. die 13. fall 14. go to … 15. graduate from 16. … happen 17. laugh 18. listen to... 19. live 20. rise 21. sit 22. smile 23. swim 24. stay (at home / in a hotel) 25. work 26. wait for 汉译英： 1.昨天我去了电影院。 2.我能用英语跟外国人自由交谈。 3.晚上7点我们到达了机场。 4.暑假就要到了。 5.现在很多老人独自居住。 6.老师同意了。 7.刚才发生了一场车祸。 8.课上我们应该认真听讲。9. 我们的态度很重要。 10. 能否成功取决于你的态度。 11. 能取得多大进步取决于你付出多少努力。 12. 这个木桶能盛多少水取决于最短的一块板子的长度。

初中英语造句

【it's time to和it's time for】 ——————这其实是一个句型,只不过后面要跟不同的东西. ——————It's time to跟的是不定式（to do）.也就是说,要跟一个动词,意思是“到做某事的时候了”.如： It's time to go home. It's time to tell him the truth. ——————It's time for 跟的是名词.也就是说,不能跟动词.如： It's time for lunch.(没必要说It's time to have lunch) It's time for class.(没必要说It's time to begin the class.) They can't wait to see you Please ask liming to study tonight. Please ask liming not to play computer games tonight. Don’t make/let me to smoke I can hear/see you dance at the stage You had better go to bed early. You had better not watch tv It’s better to go to bed early It’s best to run in the morning I am enjoy running with music. With 表伴随听音乐 I already finish studying You should keep working. You should keep on studying English Keep calm and carry on 保持冷静继续前行二战开始前英国皇家政府制造的海报名字 I have to go on studying I feel like I am flying I have to stop playing computer games and stop to go home now I forget/remember to finish my homework. I forget/remember cleaning the classroom We keep/percent/stop him from eating more chips I prefer orange to apple I prefer to walk rather than run I used to sing when I was young What’s wrong with you There have nothing to do with you I am so busy studying You are too young to na?ve I am so tired that I have to go to bed early

The Kite Runner-美句摘抄及造句

《The Kite Runner》追风筝的人--------------------------------美句摘抄 1.I can still see Hassan up on that tree, sunlight flickering through the leaves on his almost perfectly round face, a face like a Chinese doll chiseled from hardwood: his flat, broad nose and slanting, narrow eyes like bamboo leaves, eyes that looked, depending on the light, gold, green even sapphire 翻译：我依然能记得哈桑坐在树上的样子，阳光穿过叶子，照着他那浑圆的脸庞。他的脸很像木头刻成的中国娃娃，鼻子大而扁平，双眼眯斜如同竹叶，在不同光线下会显现出金色、绿色，甚至是宝石蓝。 E.g.: A shadow of disquiet flickering over his face. 2.Never told that the mirror, like shooting walnuts at the neighbor's dog, was always my idea. 翻译：从来不提镜子、用胡桃射狗其实都是我的鬼主意。E.g.:His secret died with him, for he never told anyone. 3.We would sit across from each other on a pair of high

翻译加造句

一、翻译 1. The idea of consciously seeking out a special title was new to me., but not without appeal. 让我自己挑选自己最喜欢的书籍这个有意思的想法真的对我具有吸引力。 2.I was plunged into the aching tragedy of the Holocaust, the extraordinary clash of good, represented by the one decent man, and evil. 我陷入到大屠杀悲剧的痛苦之中，一个体面的人所代表的善与恶的猛烈冲击之中。 3.I was astonished by the the great power a novel could contain. I lacked the vocabulary to translate my feelings into words. 我被这部小说所包含的巨大能量感到震惊。我无法用语言来表达我的感情（心情）。 4，make sth. long to short长话短说 5.I learned that summer that reading was not the innocent(简单的) pastime(消遣) I have assumed it to be., not a breezy, instantly forgettable escape in the hammock(吊床)，( though I’ ve enjoyed many of those too ). I discovered that a book, if it arrives at the right moment, in the proper season, will change the course of all that follows. 那年夏天，我懂得了读书不是我认为的简单的娱乐消遣，也不只是躺在吊床上，一阵风吹过就忘记的消遣。我发现如果在适宜的时间、合适的季节读一本书的话，他将能改变一个人以后的人生道路。 二、词组造句 1. on purpose 特意，故意 This is especially true here, and it was ~. (这一点在这里尤其准确，并且他是故意的) 2.think up 虚构，编造，想出 She has thought up a good idea. 她想出了一个好的主意。 His story was thought up. 他的故事是编出来的。 3. in the meantime 与此同时 助记：in advance 事前in the meantime 与此同时in place 适当地... In the meantime, what can you do? 在这期间您能做什么呢？ In the meantime, we may not know how it works, but we know that it works. 在此期间，我们不知道它是如何工作的，但我们知道，它的确在发挥作用。 4.as though 好像，仿佛 It sounds as though you enjoyed Great wall. 这听起来好像你喜欢长城。 5. plunge into 使陷入 He plunged the room into darkness by switching off the light. 他把灯一关，房