已知X正定，V为对称矩阵，则矩阵XVX是否正定，为什么

矩阵XVX一定是正定的。

首先，对于任意非零向量z，我们有z^T(XVX)z = (Xz)^TV(Xz) > 0，因为X正定，所以Xz非零，而V是对称矩阵，所以(Xz)^TV(Xz) > 0。

其次，矩阵XVX满足正定矩阵的定义：它的所有特征值都是正数。设λ是XVX的一个特征值，x是对应的特征向量，则有XVXx = λx，左右两边同时左乘x^T和x，得到λ = x^TXVXx = (Xx)^TV(Xx) > 0，因为X正定，所以Xx非零，而V是对称矩阵，所以(Xx)^TV(Xx) > 0。

综上所述，矩阵XVX是正定的。

相关问题

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对这个字符串进行解码时，使用了URLDecoder.decode，然后使用了Base64.getDecoder().decode(）方法，出现Caused by: java.lang.IllegalArgumentException: Illegal base64 character 20

这个问题的原因是因为H4sIAAAAAAAAAO1YS2/bRhD+Kwblo0XzsXxIN8F2GgN2asR2e6gLYkUuzY1ILrFcSlACA+qhaHtpDi0KFOm1l556aIuiyJ/RwS76LzrLhy3ZUmwjbZAY0kXk7Mzuzsz3fUvysxeKJzj2Bx4NlK6pIYR0yzA3FEETonR120WuYbqWriEDjOMMjEoZoGwoAc0FTX1RxioGwmZomMh3+m4Qaj444JSl44QV+VKPmPaV7gtlHf69hIiIST+fBQTG1stBpZcGnEF8ZfCGhOeUpTBgqYZqSTPOshkzUpHqNt4BEZjG5ZyJmpM0ZzwPsMAqTnE8FtTP4aqcX82Dgdrr5zI5cVh5boNn72C31Soz3hmSVOymgnCIbbUW+6rP8BC3WjqyLeVsQyEyBpZPMuFV1xtKxllGuKAkL1PPfU5I6kWEnkbg6uoa7J3ld848xbJTyv7BETqWtoAMqU+WVnw9gerKiuwX+4W873OogFztyfbTj3e3lXL1mcVsVVP1N5RZIuU5S4nHwjAnkEEbuTKFBKdFCPUpOOHg/2TnaKd3uCMj6oxHNBARYAwhbVGz5XJlEkmiyu5BCYc4l/WLsQgZT7wajlt7B2u9LIMRmnsxO6WwPcELAuiSN14Wyd2lRQK+HauDNMvQHMftVK0IPJrmFYR1XXeR3rFsG+mwgRENaTPTekrEiPFBs+anu492lXqBrZhWTe6BxceCnDI+htvHLJEwjnG/rPehwIA3yQpfVCU8wv21PuYnm9PJZDr5Yzr5cjr5cTr5azr5czp5NZ18D9eyEpCWH+E0JbHn4zjuS7rWwApxnMvtNeNN89OQwQJ6A50b1IIpQ8oh7wCP61kArV4YF3nkzXPftizTPduYEwrXtl3HdszOTaHQTe2dCQUuBDuq1/jP1YKOIhwTP2ZFILGHCrWgagQ9VfcxTXvQxCEV41b5m6X6OmARIFFuakX1B0P1dRKTBDpcbZQDFRkcBaWgXw41pq5yC6ObgHqJ5giCMp0SoT4lkDYdkj08ZoWYKWPd/7ujs2qaiOdQs5xCb6kKhmbbhmlDvW/KgmmhBywLMJCwVN0q/94sDZ9QMjos27nShwelD5yEhFcJ3o+gBb8/tu4hCkvj/2dhoAGAnl5iu74dg0iyAa1BqbsucizT7ziahdtax7YC2w1DbOu4bSHHtfp6v420TsfQ2le+vm4akla36seMOoDDs3zp49A8p3TVMFRzGc7PbtB2GewX0nkh4ocgChnj4spXvoNdmWvvatrr2SzaOPCCQLtAacOQ+l7OCg5CUCP271e/XXz788XvX/zz1csZ55xg7kfegIwB3rI6Fz/9cv7yh/Ovfz2fvPbqoG++O389mQmaQf01ElxBOxIi655snmwCJv04k9CsaJx0Xc3WTjZbJ5tx4Q+2OR59REOxB32to70My8SVk81riF1Da8dLkDlrrbBZE7fZnBexXNQ7vK6u74OMLpfLpUJ4F4V6J2J5dpdT+9pzwOXxmOFTIjGv3Hh+MFEHoD/zrNG2QC1MzZWvIF407nMaeJHVJLBSpiq1igRDmtOGCoqhGWZbs9qGuaajrul0TVO1DEPOWXlfo3Bl7HM2yoE5MU5PC+gSDD6P2n560wPeRHkpgsrx0aO2e+VwNy2qfO8hRfcBHFQnpDGB6YXHUp8swJll2HM4s1wdUONYK5Stzr8P9vxrzKsD8MM9AC1zXph0G5TGMZFjr6RpJU23S9McmGehvERC3k66FijVSmjeH6GRH6EeU3GAi8EiqUHOnNSAypjys55xL6XpnH3+L4jsCGheHAAA是经过URL编码的，其中包含了%20这个字符，这个字符在URL编码中代表的是空格。

在对这个字符串进行解码时，如果没有正确处理URL编码，就会导致解码失败并抛出IllegalArgumentException异常。

要解决这个问题，可以先对URL编码进行解码，然后再进行Base64解码。可以使用java.net.URLDecoder类的decode()方法进行URL解码，例如：

String input = "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