operations():
  Examples of linear algebra operations

  Matrix-vector multiplication: A*x=y

A
[[1 2 3]
 [4 5 6]
 [7 8 0]]
x
[1 2 3]
y = np.matmul(A,x) = A*x:
[14 32 23]

  Given A, y, solve A*x=y for x

A
[[1 2 3]
 [4 5 6]
 [7 8 0]]
y
[14 32 23]
x = np.linalg.solve(A,y)
[1. 2. 3.]
  Verify by computing A*x:
[14. 32. 23.]

  Try power method for a symmetric matrix with positive eigenvalues
0 2.982062490439092
1 4.409620360315209
2 4.417304707497886
3 4.417445579257894
4 4.417452841446047
5 4.417453318476534
6 4.4174533519433625
7 4.417453354333066
8 4.4174533545044845
9 4.417453354516794
  Estimated dominant eigenvalue is  4.417453354516794
  Estimated dominant eigenvector 
[0.5273854  0.46592293 0.54137524 0.46010142]

  np.linalg.eig() returns           4.417453354517752
  np.linalg.dominant eigenvector 
[0.52738541 0.46592283 0.54137519 0.46010157]

  Estimate matrix norm ||A|| = max ||Ax|| / ||x||

  A_norm =  13.201458618948678
  A_norm_estimate =  13.121967004653577