# Leo Zhang

A simple man with my own ideal

## 1、为什么要进行矩阵分解

个人认为，首先，当数据量很大时，将一个矩阵分解为若干个矩阵的乘积可以大大降低存储空间；其次，可以减少真正进行问题处理时的计算量，毕竟算法扫描的元素越少完成任务的速度越快，这个时候矩阵的分解是对数据的一个预处理；再次，矩阵分解可以高效和有效的解决某些问题；最后，矩阵分解可以提高算法数值稳定性，关于这一点可以有进一步的说明，

，扰动项为：

这个例子说明，当方程组常数项发生微小变动的时候会导致求出的结果差别相当大，而导致这种差别的并不是求解方法，而是方程组系数矩阵本身的问题，这会给我们解决问题带来很大危害，例如，我们在用计算机求解这类问题时难以避免在计算当中出现舍入误差，如果矩阵本身性质不好会直接导致所答非所问。

对常数向量b和矩阵A进行一个简单的扰动分析：

1)、扰动b，原方程组为：

(式子1)，（，A非奇异）

(式子2)

，可见b的扰动对解的影响由决定。

2)、扰动A，扰动后为：

(式子3)，（，A非奇异）

，整理一下就是：

，A的扰动对解的影响依然是由决定。

3)、对于同时扰动A和b的情况偶就不推了，最后的结果依然是，扰动对解的影响依然由决定。

定义矩阵的条件数来描述矩阵的病态程度，一般认为条件数小于100为良态，条件数在100到1000之间为中等程度的病态，条件数超过1000存在严重病态。以上面的矩阵A为例，采用2-范数表示的条件数为：，看来矩阵处于中等病态程度。

矩阵其实就是一个给定的线性变换，特征向量描述了这个线性变换的主要方向，而特征值描述了一个特征向量的长度在该线性变换下缩放的比例，有关特征值和特征向量的相关概念可查看http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors，对开篇的例子进一步观察发现，A是个对称正定矩阵，A的特征值分别为：14.93303437 和：0.06696563，两个特征值在数量级上相差很大，这意味着b发生扰动时，向量x在这两个特征向量方向上的移动距离是相差很大的——对于对应的特征向量只需要微小的移动就可到达b的新值，而对于，由于它比起太小了，因此需要x做大幅度移动才能到达b的新值，于是悲剧就发生了……………..。

关于矩阵可以有以下各种分解方式，①矩阵的三角分解（Cholesky分解、LU分解等），②矩阵的正交三角分解(QR分解等)，③矩阵的满秩分解，④矩阵的奇异值分解(SVD)(关于SVD可以查看高人LeftNotEasyhttp://www.cnblogs.com/LeftNotEasy/archive/2011/01/19/svd-and-applications.html#2038925一文以及他提供的参考资料)。

再看矩阵A，它是个对称正定矩阵，对这种矩阵都可以进行Cholesky分解，也就是将矩阵A分解为：，其中为一个下三角矩阵，具体操作随后讨论，回头看方程组，它就变成了：

，其中：，特征值为：2.2360680和：0.4472136

## 2、实正定Hermit矩阵的完全Cholesky分解

设矩阵A有如下形式：

，在第i次迭代时有：

，其中为i-1维单位矩阵

   1: for i:=1 to n do                                        //逐行计算L的值，w向量初始状态为实hermit矩阵A的上三角部分的一行
   2:     w(i .. n) := a(i,i .. n);
   3:     for k:=1 to i-1 do
   4:         temp := L(k,i);
   5:         if(temp != 0) then
   6:             w(i .. n) := w(i .. n) - temp*L(k,i .. n);  //计算L(i,j)公式的分子部分
   7:         endif;
   8:     endfor;
   9:     w(i) := sqrt(w(i));                                 //L矩阵对角线元素计算
  10:     w(i+1 .. n) := w(i+1 .. n) / w(i);                  //L矩阵非对角线元素计算
  11:
  12:     L(i,i .. n) := w(i .. n);                           //更新L矩阵
  13:     w(i .. n) :=0;                                      //清空w向量
  14: endfor;

   1: //串行完全Chlolesky分解,时间复杂度(O(n^3))
   2: double **complete_cholesky_decompose(double **A)
   3: {
   4:     if(NULL==A)
   5:         return NULL;
   6:
   7:     double **L=malloc_matrix();
   8:     clear_matrix(L);
   9:
  10:     int i,j,k,m;
  11:
  12:     double *w=malloc_vector();
  13:
  14:     clear_vector(w);//清除向量
  15:
  16:     for(i=0;i<LEN;i++){
  17:         for(m=i;m<LEN;m++){
  18:             w[m]=A[i][m];
  19:         }
  20:
  21:         for(k=0;k<i;k++){
  22:             double temp=L[k][i];
  23:             if(temp!=0){
  24:                 for(m=i;m<LEN;m++){
  25:                     w[m] -= temp*L[k][m];
  26:                 }
  27:             }
  28:         }
  29:
  30:         w[i]=sqrt(w[i]);
  31:         for(m=i+1;m<LEN;m++){
  32:             w[m] /=w[i];
  33:         }
  34:
  35:         for(m=i;m<LEN;m++){
  36:             L[i][m]=w[m];
  37:         }
  38:
  39:         clear_vector(w);
  40:     }
  41:
  42:     return L;
  43: }

## 3、实用工具和网址

### (1)、R

我认为R是进行关于矩阵相关操作的较好工具，它是免费的和开源的(我记的是)，功能不亚于matlab，当然它能做的事情远远不止矩阵计算，还可以做分类、聚类、回归、统计分析等等等等，R可以从http://www.r-project.org/获取到，同时有linux版本和windows版本。

1)、创建向量

   1: >  x=c(0,1,3,4,6,8,9)
   2: > x
   3:     [1] 0 1  3 4  6  8 9

   1: > x=seq(1,3,by=0.1)
   2: > x
   3: [1] 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

   1: > x=rep(0:5,rep(2,6))
   2: > x
   3: [1] 0 0 1 1 2 2 3 3 4 4 5 5

2)、创建矩阵

   1: > matrix
   2: function (data = NA, nrow = 1, ncol = 1, byrow = FALSE, dimnames = NULL)
   3: {
   4:     data <- as.vector(data)
   5:     if (missing(nrow))
   6:         nrow <- ceiling(length(data)/ncol)
   7:     else if (missing(ncol))
   8:         ncol <- ceiling(length(data)/nrow)
   9:     .Internal(matrix(data, nrow, ncol, byrow, dimnames))
  10: }
  11: <environment: namespace:base>

data存储矩阵数据，nrow和ncol分别为行数和列数，byrows默认为FALSE，表示矩阵数据生成时以列为主序，例如：

   1: > data=c(
   2: + 0.0100,0.1710,0.0967,0.1661,0.1254,0.0728,0.0928,0.1272,0.1041,0.0644,
   3: + 0.1710,3.3743,2.3896,3.5556,2.6312,1.6968,1.8502,2.6637,2.5208,1.2792,
   4: + 0.0967,2.3896,2.5098,3.0234,2.1745,1.6661,1.5959,2.5601,2.6507,1.0512,
   5: + 0.1661,3.5556,3.0234,4.3530,3.4144,2.5287,2.3244,3.6691,3.5036,1.6727,
   6: + 0.1254,2.6312,2.1745,3.4144,2.9375,2.4152,1.9473,3.0911,2.9016,1.6776,
   7: + 0.0728,1.6968,1.6661,2.5287,2.4152,2.5097,1.7161,2.8719,2.6663,2.0713,
   8: + 0.0928,1.8502,1.5959,2.3244,1.9473,1.7161,2.0160,3.2206,2.3529,1.8967,
   9: + 0.1272,2.6637,2.5601,3.6691,3.0911,2.8719,3.2206,5.8295,4.0840,3.4258,
  10: + 0.1041,2.5208,2.6507,3.5036,2.9016,2.6663,2.3529,4.0840,3.8915,2.7323,
  11: + 0.0644,1.2792,1.0512,1.6727,1.6776,2.0713,1.8967,3.4258,2.7323,4.3582
  12: + )
  13: > a=matrix(data,nrow=10,ncol=10,byrow=TRUE)
  14: > a
  15:         [,1]   [,2]   [,3]   [,4]   [,5]   [,6]   [,7]   [,8]   [,9]  [,10]
  16: [1,] 0.0100 0.1710 0.0967 0.1661 0.1254 0.0728 0.0928 0.1272 0.1041 0.0644
  17: [2,] 0.1710 3.3743 2.3896 3.5556 2.6312 1.6968 1.8502 2.6637 2.5208 1.2792
  18: [3,] 0.0967 2.3896 2.5098 3.0234 2.1745 1.6661 1.5959 2.5601 2.6507 1.0512
  19: [4,] 0.1661 3.5556 3.0234 4.3530 3.4144 2.5287 2.3244 3.6691 3.5036 1.6727
  20: [5,] 0.1254 2.6312 2.1745 3.4144 2.9375 2.4152 1.9473 3.0911 2.9016 1.6776
  21: [6,] 0.0728 1.6968 1.6661 2.5287 2.4152 2.5097 1.7161 2.8719 2.6663 2.0713
  22: [7,] 0.0928 1.8502 1.5959 2.3244 1.9473 1.7161 2.0160 3.2206 2.3529 1.8967
  23: [8,] 0.1272 2.6637 2.5601 3.6691 3.0911 2.8719 3.2206 5.8295 4.0840 3.4258
  24: [9,] 0.1041 2.5208 2.6507 3.5036 2.9016 2.6663 2.3529 4.0840 3.8915 2.7323
  25: [10,] 0.0644 1.2792 1.0512 1.6727 1.6776 2.0713 1.8967 3.4258 2.7323 4.3582

3)、矩阵转置

   1: > A=matrix(0:11,nrow=3,ncol=4,byrow=TRUE)
   2: > A
   3:      [,1] [,2] [,3] [,4]
   4: [1,]    0    1    2    3
   5: [2,]    4    5    6    7
   6: [3,]    8    9   10   11
   7: > t(A)
   8:      [,1] [,2] [,3]
   9: [1,]    0    4    8
  10: [2,]    1    5    9
  11: [3,]    2    6   10
  12: [4,]    3    7   11

4)、矩阵加减

   1: > A=matrix(0:11,nrow=3,ncol=4,byrow=TRUE)
   2: > B=matrix(1:12,nrow=3,ncol=4,byrow=TRUE)
   3: > B-A
   4:      [,1] [,2] [,3] [,4]
   5: [1,]    1    1    1    1
   6: [2,]    1    1    1    1
   7: [3,]    1    1    1    1
   8: > B+A
   9:      [,1] [,2] [,3] [,4]
  10: [1,]    1    3    5    7
  11: [2,]    9   11   13   15
  12: [3,]   17   19   21   23

5)、矩阵相乘

   1: > A=matrix(0:11,nrow=3,ncol=4,byrow=TRUE)
   2: > B=matrix(1:12,nrow=4,ncol=3,byrow=TRUE)
   3: > A%*%B
   4:      [,1] [,2] [,3]
   5: [1,]   48   54   60
   6: [2,]  136  158  180
   7: [3,]  224  262  300

6)、取方阵对角元素

   1: > A=matrix(0:8,nrow=3,ncol=3,byrow=TRUE)
   2: > A
   3:      [,1] [,2] [,3]
   4: [1,]    0    1    2
   5: [2,]    3    4    5
   6: [3,]    6    7    8
   7: > diag(A)
   8: [1] 0 4 8

7)、矩阵求逆

   1: > A=matrix(c(1,2,3,0,4,5,0,0,6),nrow=3,ncol=3,byrow=TRUE)
   2: > solve(A)
   3:      [,1]  [,2]        [,3]
   4: [1,]    1 -0.50 -0.08333333
   5: [2,]    0  0.25 -0.20833333
   6: [3,]    0  0.00  0.16666667

8)、特征值与特征向量

   1: > A=matrix(c(1,2,3,0,4,5,0,0,6),nrow=3,ncol=3,byrow=TRUE)
   2: > eigen(A)
   3: $values   4: [1] 6 4 1  5:   6:$vectors
   7:           [,1]      [,2] [,3]
   8: [1,] 0.5108407 0.5547002    1
   9: [2,] 0.7981886 0.8320503    0
  10: [3,] 0.3192754 0.0000000    0
  11:

9)、正定hermit矩阵的完全Cholesky分解

   1: > data=c(
   2: + 0.0100,0.1710,0.0967,0.1661,0.1254,0.0728,0.0928,0.1272,0.1041,0.0644,
   3: + 0.1710,3.3743,2.3896,3.5556,2.6312,1.6968,1.8502,2.6637,2.5208,1.2792,
   4: + 0.0967,2.3896,2.5098,3.0234,2.1745,1.6661,1.5959,2.5601,2.6507,1.0512,
   5: + 0.1661,3.5556,3.0234,4.3530,3.4144,2.5287,2.3244,3.6691,3.5036,1.6727,
   6: + 0.1254,2.6312,2.1745,3.4144,2.9375,2.4152,1.9473,3.0911,2.9016,1.6776,
   7: + 0.0728,1.6968,1.6661,2.5287,2.4152,2.5097,1.7161,2.8719,2.6663,2.0713,
   8: + 0.0928,1.8502,1.5959,2.3244,1.9473,1.7161,2.0160,3.2206,2.3529,1.8967,
   9: + 0.1272,2.6637,2.5601,3.6691,3.0911,2.8719,3.2206,5.8295,4.0840,3.4258,
  10: + 0.1041,2.5208,2.6507,3.5036,2.9016,2.6663,2.3529,4.0840,3.8915,2.7323,
  11: + 0.0644,1.2792,1.0512,1.6727,1.6776,2.0713,1.8967,3.4258,2.7323,4.3582
  12: + )
  13: > a=matrix(data,nrow=10,ncol=10,byrow=TRUE)
  14: > chol(a)
  15:       [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]      [,8]      [,9]     [,10]
  16: [1,]  0.1 1.7100000 0.9670000 1.6610000 1.2540000 0.7280000 0.9280000 1.2720000 1.0410000 0.6440000
  17: [2,]  0.0 0.6709694 1.0969650 1.0660545 0.7256068 0.6735329 0.3924471 0.7281703 1.1039102 0.2652282
  18: [3,]  0.0 0.0000000 0.6094086 0.4066049 0.2722586 0.3663913 0.4398089 0.8718268 0.7106926 0.2256384
  19: [4,]  0.0 0.0000000 0.0000000 0.5406286 0.8273109 0.8369753 0.3436621 0.7871389 0.5710010 0.4226980
  20: [5,]  0.0 0.0000000 0.0000000 0.0000000 0.2826847 0.7831198 0.3352446 0.2797313 0.4573777 0.9425268
  21: [6,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.2793255 0.2322540 0.9241146 0.2450380 0.8923532
  22: [7,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.7231424 1.0953043 0.3243910 0.5908204
  23: [8,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4119291 0.4303230 0.3608438
  24: [9,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.4454546 0.8321836
  25: [10,]  0.0 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.8871720

10)、矩阵奇异值分解

   1: > data=c(
   2: + 0.0100,0.1710,0.0967,0.1661,0.1254,0.0728,0.0928,0.1272,0.1041,0.0644,
   3: + 0.1710,3.3743,2.3896,3.5556,2.6312,1.6968,1.8502,2.6637,2.5208,1.2792,
   4: + 0.0967,2.3896,2.5098,3.0234,2.1745,1.6661,1.5959,2.5601,2.6507,1.0512,
   5: + 0.1661,3.5556,3.0234,4.3530,3.4144,2.5287,2.3244,3.6691,3.5036,1.6727,
   6: + 0.1254,2.6312,2.1745,3.4144,2.9375,2.4152,1.9473,3.0911,2.9016,1.6776,
   7: + 0.0728,1.6968,1.6661,2.5287,2.4152,2.5097,1.7161,2.8719,2.6663,2.0713,
   8: + 0.0928,1.8502,1.5959,2.3244,1.9473,1.7161,2.0160,3.2206,2.3529,1.8967,
   9: + 0.1272,2.6637,2.5601,3.6691,3.0911,2.8719,3.2206,5.8295,4.0840,3.4258,
  10: + 0.1041,2.5208,2.6507,3.5036,2.9016,2.6663,2.3529,4.0840,3.8915,2.7323,
  11: + 0.0644,1.2792,1.0512,1.6727,1.6776,2.0713,1.8967,3.4258,2.7323,4.3582
  12: + )
  13: > a=matrix(data,nrow=10,ncol=10,byrow=TRUE)
  14: > svd(a)
  15: $d  16: [1] 2.419118e+01 4.155429e+00 1.305297e+00 1.042439e+00 7.271958e-01 1.724779e-01 1.188910e-01 7.550273e-02 6.811257e-04 4.025360e-04  17:   18:$u
  19:              [,1]        [,2]        [,3]          [,4]        [,5]         [,6]        [,7]        [,8]         [,9]        [,10]
  20:  [1,] -0.01423809 -0.01770852  0.01046421  0.0440060554 -0.03708011  0.003989202  0.02212364 -0.02965763  0.609656748 -0.789301225
  21:  [2,] -0.30655677 -0.38844316  0.21540302  0.5860837223 -0.20831346 -0.071321462  0.09280202 -0.53705509  0.058902791  0.127479773
  22:  [3,] -0.27500938 -0.28653235 -0.04046986 -0.0006133085  0.62767935 -0.424262650 -0.38187184  0.16164557  0.260582126  0.163683441
  23:  [4,] -0.39289768 -0.37181967  0.10218229  0.0430729100 -0.04830660  0.258172491 -0.19145885  0.37788201 -0.530454225 -0.406528598
  24:  [5,] -0.32434239 -0.18416219  0.18854340 -0.2694300561 -0.38073924  0.273128771  0.08003428  0.37039768  0.491516620  0.384701079
  25:  [6,] -0.28065826  0.08302171  0.24277466 -0.6198550162 -0.27053693 -0.402153630 -0.20235690 -0.39991126 -0.144858619 -0.119997665
  26:  [7,] -0.26573534  0.09522360 -0.29346909  0.1374724777 -0.21528535 -0.600459122  0.52653396  0.34671820 -0.089612967 -0.053976475
  27:  [8,] -0.44372190  0.30810327 -0.69462776  0.0748652314 -0.12439310  0.238340862 -0.32629727 -0.18852619  0.059652991  0.047118889
  28:  [9,] -0.38271998  0.06649679  0.03894330 -0.2531095648  0.52173933  0.307376912  0.59639765 -0.24228533 -0.036358521 -0.033403328
  29: [10,] -0.27906174  0.69225888  0.52609547  0.3276170825  0.08203487 -0.008974789 -0.14718647  0.17374370  0.008992724  0.007135959
  30:
  31: $v  32: [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]  33: [1,] -0.01423809 -0.01770852 0.01046421 0.0440060554 -0.03708011 0.003989202 0.02212364 -0.02965763 0.609656748 -0.789301225  34: [2,] -0.30655677 -0.38844316 0.21540302 0.5860837223 -0.20831346 -0.071321462 0.09280202 -0.53705509 0.058902791 0.127479773  35: [3,] -0.27500938 -0.28653235 -0.04046986 -0.0006133085 0.62767935 -0.424262650 -0.38187184 0.16164557 0.260582126 0.163683441  36: [4,] -0.39289768 -0.37181967 0.10218229 0.0430729100 -0.04830660 0.258172491 -0.19145885 0.37788201 -0.530454225 -0.406528598  37: [5,] -0.32434239 -0.18416219 0.18854340 -0.2694300561 -0.38073924 0.273128771 0.08003428 0.37039768 0.491516620 0.384701079  38: [6,] -0.28065826 0.08302171 0.24277466 -0.6198550162 -0.27053693 -0.402153630 -0.20235690 -0.39991126 -0.144858619 -0.119997665  39: [7,] -0.26573534 0.09522360 -0.29346909 0.1374724777 -0.21528535 -0.600459122 0.52653396 0.34671820 -0.089612967 -0.053976475  40: [8,] -0.44372190 0.30810327 -0.69462776 0.0748652314 -0.12439310 0.238340862 -0.32629727 -0.18852619 0.059652991 0.047118889  41: [9,] -0.38271998 0.06649679 0.03894330 -0.2531095648 0.52173933 0.307376912 0.59639765 -0.24228533 -0.036358521 -0.033403328  42: [10,] -0.27906174 0.69225888 0.52609547 0.3276170825 0.08203487 -0.008974789 -0.14718647 0.17374370 0.008992724 0.007135959 11)、矩阵QR分解  1: > A=matrix(1:12,3,4)  2: > qr(A)  3:$qr
   4:            [,1]      [,2]          [,3]          [,4]
   5: [1,] -3.7416574 -8.552360 -1.336306e+01 -1.817376e+01
   6: [2,]  0.5345225  1.963961  3.927922e+00  5.891883e+00
   7: [3,]  0.8017837  0.988693  3.443426e-16 -3.439089e-16
   8:
   9: $rank  10: [1] 2  11:   12:$qraux
  13: [1] 1.267261e+00 1.149954e+00 3.443426e-16 3.439089e-16
  14:
  15: \$pivot
  16: [1] 1 2 3 4
  17:
  18: attr(,"class")
  19: [1] "qr"

### (2)、找论文的好去处

1)、http://www.pdfgratis.com/一个非常强大的网站，能下载到各种论文；

### (3)、编写latex公式的工具

最初是在小桥流水的博客上发现的，地址如下：http://www.cnblogs.com/youwang/archive/2010/04/04/1704099.html，对他的插件做了一点修改，增加的功能是：可以还原公式，例如：用该工具写公式， ，选中该公式，然后单击插件，可以看到公式的latex代码，源码可以在这里下载到。

posted on 2011-03-22 15:28  Leo Zhang  阅读(19080)  评论(21编辑  收藏