#! /usr/bin/env python3
#
def about_a_matrix ( ):

## about_a_matrix() reports some facts about a matrix.
#
  import numpy as np

  print ( '' )
  print ( 'about_a_matrix():' )
  print ( '  Linear algebra questions about a matrix.' )
#
#  Define M = matrix, a 2D numpy array.
#
  M = np.array ( [
    [ 1, 2,  3 ],
    [ 4, 5,  6 ],
    [ 7, 8, 10 ] ] )
  print ( '' )
  print ( '  Matrix M:' )
  print ( M )
#
#  Define D = matrix, a 2D numpy array.
#
  from dif2 import dif2

  A = dif2 ( 6 )

  print ( '' )
  print ( '  Matrix A:' )
  print ( A )
  print ( '' )
  print ( '  Python properties of A:' )
  print ( '' )
  print ( '  type(A) = ', type ( A ) )
  print ( '  len(A) is not the size, but the number of rows!' )
  print ( '  len(A) = ', len ( A ) )
  print ( '' )
  print ( '  Numpy properties of A:' )
  print ( '' )
  print ( '  A.ndim = ', A.ndim )
  print ( '  A.shape = ', A.shape )
  print ( '  A.size is the size or number of elements.' )
  print ( '  A.size = ', A.size )
  print ( '  A.dtype = ', A.dtype )
#
#  Transpose:
#
  print ( '' )
  print ( '  Transpose is A.T or np.transpose(A)' )
  print ( A.T )
#
#  Flipud:
#
  print ( '' )
  print ( '  Flip up and down = np.flipud(A)' )
  print ( np.flipud ( A ) )
#
#  Inverse:
#
  print ( '' )
  print ( '  Inverse is np.linalg.inv(A)' )
  print ( np.linalg.inv(A) )
#
#  Determinant (of a square matrix):
#
  print ( '' )
  print ( '  Determinant is np.linalg.det(A)' )
  print ( np.linalg.det(A) )
#
#  Condition (of a square matrix):
#
  print ( '' )
  print ( '  Condition is np.linalg.cond(A)' )
  print ( np.linalg.cond(A) )
#
#  Trace:
#
  print ( '' )
  print ( '  Trace is np.trace(A)' )
  print ( np.trace(A) )
#
#  Get diagonal of a matrix:
#
  print ( '' )
  print ( '  d = np.diag(A) returns diagonal vector.' )
  d = np.diag ( A )
  print ( d )
#
#  Create diagonal matrix from vector.
#
  print ( '' )
  print ( '  B = np.diag(v) creates diagonal matrix from vector.' )
  v = np.array ( [ 1, 2, 3 ] )
  B = np.diag ( v )
  print ( B )
#
#  Create a diagonal matrix from the diagonal of another matrix.
#
  print ( '' )
  print ( '  C = np.diag ( np.diag ( M ) ) ) uses diagonal of M' )
  print ( '  to create new diagonal matrix C' )
  C = np.diag ( np.diag ( M ) )
  print ( C ) 
#
#  tril(M,k) creates a lower triangular matrix
#
  print ( '' )
  print ( '  L = np.tril(M,k) creates lower triangular matrix' )
  print ( '  starting at diagonal k.' )
  L = np.tril ( M, - 1 )
  print ( L )
#
#  triu(M,k) creates an upper triangular matrix
#
  print ( '' )
  print ( '  U = np.triu(M,k) creates upper triangular matrix' )
  print ( '  starting at diagonal k' )
  U = np.triu ( M, 0 )
  print ( U )
#
#  Spy displays a plot of 0 / nonzero entries.
#
  print ( '' )
  print ( '  np.spy(A) displays 0/nonzero entries' )
  import matplotlib.pyplot as plt
  from dif2 import dif2
  A10 = dif2 ( 10 )
  plt.spy ( A10, marker = 'o' )
  plt.grid ( True )
  plt.savefig ( 'dif2_spy.png' )
  plt.show ( )
#
#  Row sums:
#
  print ( '' )
  print ( '  Row sums is np.sum(M) or np.sum(M,axis=1)' )
  print ( np.sum(M) )
  print ( np.sum(M,axis=1) )
#
#  Column sums:
#
  print ( '' )
  print ( '  Column sums is np.sum(M,axis=0)' )
  print ( np.sum(M,axis=0) )
#
#  Matrix sum
#
  print ( '' )
  print ( '  Matrix sum np.sum(np.sum(M))' )
  print ( np.sum(np.sum(M)) )
#
#  Matrix norms.
#
  A_norm = np.linalg.norm ( A )
  A_norm_one = np.linalg.norm ( A, 1 ) # L1 norm: maximum column sum of |A|
  A_norm_two = np.linalg.norm ( A, 2 ) # L2 norm: maximum eigenvalue of A'A
  A_norm_inf = np.linalg.norm ( A, np.inf ) # Linfinity norm: maximum row sum of |A|
  A_norm_fro = np.linalg.norm ( A, 'fro' ) # Frobenius: sqrt of sum of squares of entries
  print ( '' )
  print ( '  Matrix norms (2 is default, fro is frobenius:' )
  print ( '' )
  print ( '  np.linalg.norm(A) = ', A_norm )
  print ( '  np.linalg.norm(A,1) = ', A_norm_one )
  print ( '  np.linalg.norm(A,2) = ', A_norm_two )
  print ( '  np.linalg.norm(A,np.inf) = ', A_norm_inf )
  print ( '  np.linalg.norm(A,\'fro\') = ', A_norm_fro )
#
#  Row 2, Column 1
#
  print ( '' )
  print ( '  Row 2, Column 1: M[2,1]' )
  print ( M[2,1] )
#
#  Row 2
#
  print ( '' )
  print ( '  Row 2: M[2,:]' )
  print ( M[2,:] )
#
#  Column 1
#
  print ( '' )
  print ( '  Column 1: M[:,1]' )
  print ( M[:,1] )
  return

if ( __name__ == "__main__" ):
  about_a_matrix ( )