about_a_matrix():
  Linear algebra questions about a matrix.

  Matrix M:
[[ 1  2  3]
 [ 4  5  6]
 [ 7  8 10]]

  Matrix A:
[[ 2. -1.  0.  0.  0.  0.]
 [-1.  2. -1.  0.  0.  0.]
 [ 0. -1.  2. -1.  0.  0.]
 [ 0.  0. -1.  2. -1.  0.]
 [ 0.  0.  0. -1.  2. -1.]
 [ 0.  0.  0.  0. -1.  2.]]

  Python properties of A:

  type(A) =  <class 'numpy.ndarray'>
  len(A) is not the size, but the number of rows!
  len(A) =  6

  Numpy properties of A:

  A.ndim =  2
  A.shape =  (6, 6)
  A.size is the size or number of elements.
  A.size =  36
  A.dtype =  float64

  Transpose is A.T or np.transpose(A)
[[ 2. -1.  0.  0.  0.  0.]
 [-1.  2. -1.  0.  0.  0.]
 [ 0. -1.  2. -1.  0.  0.]
 [ 0.  0. -1.  2. -1.  0.]
 [ 0.  0.  0. -1.  2. -1.]
 [ 0.  0.  0.  0. -1.  2.]]

  Flip up and down = np.flipud(A)
[[ 0.  0.  0.  0. -1.  2.]
 [ 0.  0.  0. -1.  2. -1.]
 [ 0.  0. -1.  2. -1.  0.]
 [ 0. -1.  2. -1.  0.  0.]
 [-1.  2. -1.  0.  0.  0.]
 [ 2. -1.  0.  0.  0.  0.]]

  Inverse is np.linalg.inv(A)
[[0.85714286 0.71428571 0.57142857 0.42857143 0.28571429 0.14285714]
 [0.71428571 1.42857143 1.14285714 0.85714286 0.57142857 0.28571429]
 [0.57142857 1.14285714 1.71428571 1.28571429 0.85714286 0.42857143]
 [0.42857143 0.85714286 1.28571429 1.71428571 1.14285714 0.57142857]
 [0.28571429 0.57142857 0.85714286 1.14285714 1.42857143 0.71428571]
 [0.14285714 0.28571429 0.42857143 0.57142857 0.71428571 0.85714286]]

  Determinant is np.linalg.det(A)
6.999999999999998

  Condition is np.linalg.cond(A)
19.195669358089216

  Trace is np.trace(A)
12.0

  d = np.diag(A) returns diagonal vector.
[2. 2. 2. 2. 2. 2.]

  B = np.diag(v) creates diagonal matrix from vector.
[[1 0 0]
 [0 2 0]
 [0 0 3]]

  C = np.diag ( np.diag ( M ) ) ) uses diagonal of M
  to create new diagonal matrix C
[[ 1  0  0]
 [ 0  5  0]
 [ 0  0 10]]

  L = np.tril(M,k) creates lower triangular matrix
  starting at diagonal k.
[[0 0 0]
 [4 0 0]
 [7 8 0]]

  U = np.triu(M,k) creates upper triangular matrix
  starting at diagonal k
[[ 1  2  3]
 [ 0  5  6]
 [ 0  0 10]]

  np.spy(A) displays 0/nonzero entries

  Row sums is np.sum(M) or np.sum(M,axis=1)
46
[ 6 15 25]

  Column sums is np.sum(M,axis=0)
[12 15 19]

  Matrix sum np.sum(np.sum(M))
46

  Matrix norms (2 is default, fro is frobenius:

  np.linalg.norm(A) =  5.830951894845301
  np.linalg.norm(A,1) =  4.0
  np.linalg.norm(A,2) =  3.801937735804838
  np.linalg.norm(A,np.inf) =  4.0
  np.linalg.norm(A,'fro') =  5.830951894845301

  Row 2, Column 1: M[2,1]
8

  Row 2: M[2,:]
[ 7  8 10]

  Column 1: M[:,1]
[2 5 8]