【正文】 s Day! in shops, hotels and restaurants. But few young women will mark the festival with their boyfriends, or take part in traditional activities to pray for cleverness.。s Day on February 14, characterized by bouquets of roses, chocolates and romantic candlelight dinners, than they do about their homegrown day for lovers. Even Qixi is nowadays referred to as the Chinese Valentine39。s 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 BCAD 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 Daughters39。s 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 itShutterbugs 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. You cannot miss out the Autumn of Arxan. It is definitely the best with brightlycolored scenery full of emotions. Nestled close to the country39。ll Hermite matrix0 前言矩陣的和與乘積是矩陣的兩種基本運算，它們的特征值、秩、正定性等方面的關系問題,在理論上和實際應用中都很有意義,例如 矩陣特征值與奇異值估計在矩陣計算、誤差分析中有著重要的應用, ,的乘積,一般主要研究它們的可交換性. 但事實上, 矩陣對 ,它們的和與積相等. 這對矩陣在矩陣的秩、特征值和特征向量、正定性、為矩陣的秩，表示矩陣的轉置，為階Hermite 矩陣，為矩陣的跡，表示矩陣的共軛轉置，：，即階矩陣對符合條件.如矩陣和以及和都是符合條件的矩陣對.1 引理及相關定理定義 設且，若，有，則為正定矩陣.定義 設，若，則稱為規(guī)范矩陣.引理 若是正定矩陣，則.引理2 若，是非奇異矩陣，則是正定矩陣的充要條件是是正定矩陣.引理 是規(guī)范矩陣，若，則是正定矩陣.引理4 相似的矩陣有相同的特征值.引理5 階矩陣，符合條件的充分必要條件是和互為逆矩陣；且若矩陣對符合條件，則及 證明 因為 .即. 又和互為逆矩陣，所以，故.引理6 若矩陣對符合條件，則存在階非奇異矩陣和，使得.證明 由引理1顯然得證.引理 （Hoffman—wieland定理）設，均為實對稱陣，它們的特征值分別為：，則，的特征值之間有如下關系成立：引理 （Neumann 不等式）設，的特征值分別為，則 （1）設，的奇異值分別為，則 （2）引理 設是交換族，那么存在一個酉陣，使得對每個，是上三角的.定理1 設、是正定對稱矩陣，則是正定矩陣的充要條件是.證明 若是正定矩陣，由引理2知，是正定矩陣,，若，則由引理4得.因 故 因此，是規(guī)范矩陣，由引理3知，.定理2 若矩陣對符合條件，則（1）矩陣和的特征值均不為1；若是的特征值，則對應的特征，和有公共的特征向量系；（2）可以對角化的充分必要條件是可以對角化，即，可以同時對角化；（3）若有個不同的特征值，存在一個次數(shù)不超過的多項式使得,證明 （1）由引理1，即1既不是的特征值，對應的特征值是,則，而,故,，所以也是矩陣的特征向量，對應的特征值為 ；若是矩陣的特征向量，同理可證它也是的特征向量,這說明與有公共的特征向量系.(2) ,因而存在非奇異矩陣,使,為的特征值,所以令則, 為的特征向量,由(1) 知, 也是矩陣B的特征向量,設,為的特征值, ,則,于是相似于對角矩陣.(3) 有個不同的特征值,故可對角化,由(2) 知也可對角化. 令,取多項式,由于互不相同,根據(jù) Lagrange 插值定理可知,存在一個次數(shù)不超過的多項式 ,使得 ,則 ,即有,從而 ,定理1得證.推論 設,為正定的Hermite 陣，且滿足條件，則存在酉陣，使得和同時為對角陣.定理3 若,都是數(shù)域上滿足條件的矩陣，若,的特征值都在中，則存在上非奇異矩陣，使得及都是上三角矩陣，即，可同時上三角化.證明 ，結論顯然，假定對階矩陣結論成立，因為，滿足條件，則，且與