def symm_random_eigen_right ( n, d, key ): #*****************************************************************************80 # ## SYMM_RANDOM_EIGEN_RIGHT returns right eigenvectors for the SYMM_RANDOM matrix. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 11 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, integer N, the order of A. # # Input, real D(N), the desired eigenvalues for the matrix. # # Input, integer KEY, a positive integer that selects the data. # # Output, real V(N,N), the vectors. # from orth_random import orth_random # # Get a random orthogonal matrix Q. # x = orth_random ( n, key ) return x
def symm_random ( n, d, key ):
#*****************************************************************************80
#
## SYMM_RANDOM returns the SYMM_RANDOM matrix.
#
# Properties:
#
# A is symmetric: A' = A.
#
# The eigenvalues of A will be real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 11 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, real D(N), the desired eigenvalues for the matrix.
#
# Input, integer KEY, a positive integer that selects the data.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from orth_random import orth_random
#
# Get a random orthogonal matrix Q.
#
q = orth_random ( n, key )
#
# Set A = Q * Lambda * Q'.
#
a = np.zeros ( ( n, n ) )
for j in range ( 0, n ):
for i in range ( 0, n ):
for k in range ( 0, n ):
a[i,j] = a[i,j] + q[i,k] * d[k] * q[j,k]
return a
def idem_random_eigen_right ( n, rank, key ): #*****************************************************************************80 # ## IDEM_RANDOM_EIGEN_RIGHT returns the right eigenvectors of the IDEM_RANDOM matrix. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 17 March 2015 # # Author: # # John Burkardt # # Reference: # # Alston Householder, John Carpenter, # The singular values of involutory and of idempotent matrices, # Numerische Mathematik, # Volume 5, 1963, pages 234-237. # # Parameters: # # Input, integer N, the order of the matrix. # # Input, integer RANK, the rank of A. # 0 <= RANK <= N. # # Input, integer KEY, a positive value that selects the data. # # Output, real X(N,N), the matrix. # import numpy as np from orth_random import orth_random x = orth_random ( n, key ) x = np.transpose ( x ) return x
def pds_random_eigen_right ( n, key ): #*****************************************************************************80 # ## PDS_RANDOM_EIGEN_RIGHT returns the right eigenvectors of the PDS_RANDOM matrix. # # Licensing: # # This code is distributed under the GNU LGPL license. # # Modified: # # 20 March 2015 # # Author: # # John Burkardt # # Parameters: # # Input, integer N, the order of A. # # Input, integer KEY, a positive value that selects the data. # # Output, real Q(N,N), the matrix. # from orth_random import orth_random from r8vec_uniform_01 import r8vec_uniform_01 # # Get a random set of eigenvalues. # seed = key lam, seed = r8vec_uniform_01 ( n, seed ) # # Get a random orthogonal matrix Q. # q = orth_random ( n, key ) return q
def pds_random_eigen_right(n, key):
#*****************************************************************************80
#
## PDS_RANDOM_EIGEN_RIGHT returns the right eigenvectors of the PDS_RANDOM matrix.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 20 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, integer KEY, a positive value that selects the data.
#
# Output, real Q(N,N), the matrix.
#
from orth_random import orth_random
from r8vec_uniform_01 import r8vec_uniform_01
#
# Get a random set of eigenvalues.
#
seed = key
lam, seed = r8vec_uniform_01(n, seed)
#
# Get a random orthogonal matrix Q.
#
q = orth_random(n, key)
return q
def pds_random ( n, key ):
#*****************************************************************************80
#
## PDS_RANDOM returns a random positive definite symmetric matrix.
#
# Discussion:
#
# The matrix returned will have eigenvalues in the range [0,1].
#
# Properties:
#
# A is symmetric: A' = A.
#
# A is positive definite: 0 < x'*A*x for nonzero x.
#
# The eigenvalues of A will be real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 20 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, integer KEY, a positive value that selects the data.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from orth_random import orth_random
from r8vec_uniform_01 import r8vec_uniform_01
#
# Get a random set of eigenvalues.
#
seed = key
lam, seed = r8vec_uniform_01 ( n, seed )
#
# Get a random orthogonal matrix Q.
#
q = orth_random ( n, key )
#
# Set A = Q * Lambda * Q'.
#
a = np.zeros ( ( n, n ) )
for i in range ( 0, n ):
for j in range ( 0, n ):
for k in range ( 0, n ):
a[i,j] = a[i,j] + q[i,k] * lam[k] * q[j,k]
return a
def pds_random(n, key):
#*****************************************************************************80
#
## PDS_RANDOM returns a random positive definite symmetric matrix.
#
# Discussion:
#
# The matrix returned will have eigenvalues in the range [0,1].
#
# Properties:
#
# A is symmetric: A' = A.
#
# A is positive definite: 0 < x'*A*x for nonzero x.
#
# The eigenvalues of A will be real.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 20 March 2015
#
# Author:
#
# John Burkardt
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, integer KEY, a positive value that selects the data.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from orth_random import orth_random
from r8vec_uniform_01 import r8vec_uniform_01
#
# Get a random set of eigenvalues.
#
seed = key
lam, seed = r8vec_uniform_01(n, seed)
#
# Get a random orthogonal matrix Q.
#
q = orth_random(n, key)
#
# Set A = Q * Lambda * Q'.
#
a = np.zeros((n, n))
for i in range(0, n):
for j in range(0, n):
for k in range(0, n):
a[i, j] = a[i, j] + q[i, k] * lam[k] * q[j, k]
return a
def idem_random ( n, rank, key ):
#*****************************************************************************80
#
## IDEM_RANDOM returns a random idempotent matrix of rank K.
#
# Properties:
#
# A is idempotent: A * A = A
#
# If A is invertible, then A must be the identity matrix.
# In other words, unless A is the identity matrix, it is singular.
#
# I - A is also idempotent.
#
# All eigenvalues of A are either 0 or 1.
#
# rank(A) = trace(A)
#
# A is a projector matrix.
#
# The matrix I - 2A is involutory, that is ( I - 2A ) * ( I - 2A ) = I.
#
# Licensing:
#
# This code is distributed under the GNU LGPL license.
#
# Modified:
#
# 14 February 2015
#
# Author:
#
# John Burkardt
#
# Reference:
#
# Alston Householder, John Carpenter,
# The singular values of involutory and of idempotent matrices,
# Numerische Mathematik,
# Volume 5, 1963, pages 234-237.
#
# Parameters:
#
# Input, integer N, the order of A.
#
# Input, integer RANK, the rank of A.
# 0 <= RANK <= N.
#
# Input, integer KEY, a positive value that selects the data.
#
# Output, real A(N,N), the matrix.
#
import numpy as np
from orth_random import orth_random
from sys import exit
if ( rank < 0 or n < rank ):
print ''
print 'IDEM_RANDOM - Fatal error!'
print ' RANK must be between 0 and N.'
print ' Input value = %d' % ( rank )
exit ( 'IDEM_RANDOM - Fatal error!' )
#
# Get a random orthogonal matrix Q.
#
q = orth_random ( n, key )
#
# Compute Q' * D * Q, where D is the diagonal eigenvalue matrix
# of RANK 1's followed by N-RANK 0's.
#
a = np.zeros ( ( n, n ) )
for i in range ( 0, n ):
for j in range ( 0, n ):
t = 0.0
for k in range ( 0, rank ):
t = t + q[k,i] * q[k,j]
a[i,j] = t
return a