python实现svd分解

svd分解：

中间的矩阵为对角矩阵
python实现：

from numpy import mat, linalg


def main():
    data = mat([[1, 1, 1, 0, 0],
                [2, 2, 2, 0, 0],
                [1, 1, 1, 0, 0],
                [5, 5, 5, 0, 0],
                [1, 1, 0, 2, 2],
                [0, 0, 0, 3, 3],
                [0, 0, 0, 1, 1]])
    u, sigma, v_t = linalg.svd(data)
    print(u)
    print()
    print(sigma)
    print()
    print(v_t)
    sigma_mat = mat([[sigma[0], 0, 0], [0, sigma[1], 0], [0, 0, sigma[2]]])
    new_data = u[:, :3] * sigma_mat * v_t[:3, :]
    print()
    print(new_data)


if __name__ == '__main__':
    main()

输出：

[[-1.77939726e-01 -1.64228493e-02  1.80501685e-02  9.27047702e-01
  3.74969832e-02 -3.11055905e-01 -1.00810187e-01]
[-3.55879451e-01 -3.28456986e-02  3.61003369e-02  3.29089057e-02
 -9.11915822e-01  1.95688838e-01 -3.96847175e-04]
[-1.77939726e-01 -1.64228493e-02  1.80501685e-02  2.74228495e-01
  2.65720721e-01  8.63025454e-01  2.77705297e-01]
[-8.89698628e-01 -8.21142464e-02  9.02508423e-02 -2.53418802e-01
  3.04122788e-01 -1.88669445e-01 -3.52202831e-02]
[-1.33954753e-01  5.33527340e-01 -8.35107599e-01  5.55111512e-16
 -1.58727198e-16 -9.22872889e-16 -3.34801631e-16]
[-2.15749771e-02  7.97677135e-01  5.13074760e-01 -2.77858188e-03
  1.97619258e-02  9.19461850e-02 -3.01906682e-01]
[-7.19165903e-03  2.65892378e-01  1.71024920e-01  8.33574565e-03
 -5.92857774e-02 -2.75838555e-01  9.05720047e-01]]

[9.72140007e+00 5.29397912e+00 6.84226362e-01 1.33460130e-15
2.63935250e-31]

[[-5.81200877e-01 -5.81200877e-01 -5.67421508e-01 -3.49564973e-02
 -3.49564973e-02]
[ 4.61260083e-03  4.61260083e-03 -9.61674228e-02  7.03814349e-01
  7.03814349e-01]
[-4.02721076e-01 -4.02721076e-01  8.17792552e-01  5.85098794e-02
  5.85098794e-02]
[-7.07106781e-01  7.07106781e-01  1.66533454e-16  3.46944695e-17
  6.93889390e-18]
[ 0.00000000e+00  2.60253608e-17 -2.36845203e-17 -7.07106781e-01
  7.07106781e-01]]

[[ 1.00000000e+00  1.00000000e+00  1.00000000e+00  1.94093903e-16
  1.74795104e-16]
[ 2.00000000e+00  2.00000000e+00  2.00000000e+00  6.25868351e-17
  2.39892377e-17]
[ 1.00000000e+00  1.00000000e+00  1.00000000e+00 -6.28062537e-16
 -6.47361335e-16]
[ 5.00000000e+00  5.00000000e+00  5.00000000e+00  1.14233797e-16
  1.75229637e-17]
[ 1.00000000e+00  1.00000000e+00 -1.80048454e-16  2.00000000e+00
  2.00000000e+00]
[-5.06382306e-17  3.15500161e-16 -4.14592885e-16  3.00000000e+00
  3.00000000e+00]
[-1.01278813e-18  1.18720378e-16 -1.89063042e-16  1.00000000e+00
  1.00000000e+00]]

发现第二个矩阵（奇异值）中前3比较大，后2都是极小值，因此可以使用前三进行重构，得到接近原矩阵的矩阵