高等数学拾遗01--------Hesse矩阵与多元函数的极值

• 鞍点

  •  在处的Hesse矩阵是不定的

• 极值点
  • 是一个极小值点，那么在处的Hesse矩阵是半正定的
  • 是一个极大值点，那么在处的Hesse矩阵是半负定的
  • 在处的Hesse矩阵是正定的， 是一个严格的极小值点

  • 在处的Hesse矩阵是负定的， 是一个严格的极大值点

Hesse矩阵

• 在点处的泰勒展开式的矩阵形式

Numpy 求矩阵的特征值并做判断

• import numpy as np
  a = [[1,0,0],[-2,5,-2],[-2,4,-1]]
  c = np.linalg.eig(a)
  eig_values = c[0]
  p = list(c[0])
  positive = 0 
  negative = 0
  null = 0
  
  for i in p:
      if i > 0:
          positive += 1
      elif i == 0"
          null += 1
      elif i < 0:
          negative += 1       
  if negative > 0 and positive > 0:
      print("it is a saddle point")
  elif negative > 0 and positive = 0 and null > 0:
      print("it is a negative semidefinite matrix")
      print("Warning!!cannot judge!!")
  elif negative > 0 and positive = 0 and null = 0:
      print("it is a maximum point")
  elif negative = 0 and positive > 0 and null = 0:
      print("it is a minmum point")
  elif negative = 0 and positive > 0 and null > 0:
      print("it is a positive semidefinite matrix")
      print("Warning!!cannot judge!!")