la4j 没有正确计算矩阵的 SVD？答案

【问题标题】：la4j not calculating SVD of matrix correctly?la4j 没有正确计算矩阵的 SVD？
【发布时间】：2018-05-23 04:33:38
【问题描述】：

我有以下矩阵：

 0.003,0.013,0.022,0.013,0.003
 0.013,0.060,0.098,0.060,0.013
 0.022,0.098,0.162,0.098,0.022
 0.013,0.060,0.098,0.060,0.013
 0.003,0.013,0.022,0.013,0.003

我正在尝试使用 la4j 计算其奇异值分解，使用以下代码：

  SingularValueDecompositor SVD = new SingularValueDecompositor(A);
  Matrix[] factorization = SVD.decompose();

  Matrix U = factorization[0];
  Matrix D = factorization[1];
  Matrix V = factorization[2];

然而， U、D、V 的结果依次为：

0.102,-0.826, 0.307,-0.456,-0.071
0.456,-0.175,-0.859,-0.155,-0.017
0.751, 0.423, 0.374,-0.337,-0.059
0.456,-0.320, 0.166, 0.740, 0.339
0.102,-0.076, 0.029, 0.327,-0.936

-----------
.2873094460,.0000000000,.0000000000,.0000000000,.0000000000
.0000000000,.0000000000,.0000000000,.0000000000,.0000000000
.0000000000,.0000000000,.0000000000,.0000000000,.0000000000
.0000000000,.0000000000,.0000000000,.0000000000,.0000000000
.0000000000,.0000000000,.0000000000,.0000000000,.0000000000

-----------
0.102,-0.051, 0.975,-0.148,-0.122
0.456, 0.870, 0.030, 0.186, 0.027
0.751,-0.481,-0.015, 0.374, 0.253
0.456,-0.079,-0.221,-0.627,-0.586
0.102, 0.061,-0.009,-0.640, 0.759

这三个矩阵相乘成

0.003,-0.001,0.028,-0.004,-0.004
0.013,-0.007,0.128,-0.019,-0.016
0.022,-0.011,0.210,-0.032,-0.026
0.013,-0.007,0.128,-0.019,-0.016
0.003,-0.001,0.028,-0.004,-0.004

这不是A。我认为部分问题可能是实际对角线奇异值小于0.001（第一个除外），所以它们没有出现。（小数是 10 位的原因是因为我将十进制格式设置为）。我的问题是，如何解决这个问题并让所有奇异值都显示出来？

【问题讨论】：

标签： java linear-algebra la4j

【解决方案1】：

奇异值分解尝试将矩阵A 分解为A = U * D * V'，其中V' 是V 的转置。

// Instead of 
A = U.multiply(D.multiply(V));

// Try
A = U.multiply(D.multiply(V.transpose()));

// You should get back:
0.0030    0.0134    0.0220    0.0134    0.0030
0.0134    0.0597    0.0984    0.0597    0.0134
0.0220    0.0984    0.1620    0.0984    0.0220
0.0134    0.0597    0.0984    0.0597    0.0134
0.0030    0.0134    0.0220    0.0134    0.0030

【讨论】：