Software tutorial/Matrix operations

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LU decomposition

Let \(A = \left(\begin{array}{cccc} 4 & 0 & 1 \\ 0 & 6 & 2 \\ 3.2 & 7 & 4 \end{array} \right)\) and let \(b = \left(\begin{array}{cccc} 9.1 \\ 2 \\ 0 \end{array} \right) \)

MATLAB Python
A = [4, 0, 1; 0, 6, 2; 3.2, 7, 4];

% LU decomposition of A
[L,U,P] = lu(A)
L =
    1.0000         0         0
    0.8000    1.0000         0
         0    0.8571    1.0000
U =
    4.0000         0    1.0000
         0    7.0000    3.2000
         0         0   -0.7429
P =
         1         0         0
         0         0         1
         0         1         0

The P matrix shows that rows 2 and rows 3 were interchanged during the Gauss elimination pivoting steps.

import numpy as np
from numpy.linalg import *
from scipy.linalg import *

A = np.array([[4, 0, 1],[0, 6, 2],[3.2, 7, 4]])
LU, P = lu_factor(A)

>>> LU
array([[ 4.        ,  0.        ,  1.        ],
       [ 0.8       ,  7.        ,  3.2       ],
       [ 0.        ,  0.85714286, -0.74285714]])

>>> P
array([0, 2, 2])

The LU matrices are overlayed in the output. Notice that the results are the same as MATLAB's, except we do not need to store the diagonal set of ones from the L matrix.

The P vector is also a bit cryptic - it should be read from left to right, in order. The way to read it is that row \(i\) of matrix \(A\) was interchanged with row P[i] in \(A\), where i is the index into vector P.

  • row 0 in \(A\) was interchanged with row 0: i.e. no pivoting for the first row
  • row 1 in \(A\) was interchanged with row 2: i.e. rows 2 and 3 in \(A\) were swapped
  • row 2 in \(A\) was interchanged with row 2: i.e. no further pivoting.

Once we have \(L, U\) and \(P\) we can solve the system of equations for a given vector, \(b\). It takes a bit more work to do this in MATLAB than with Python. Recall from class that \(A = LU\), but with a permutation or pivot matrix, it becomes \(PA = LU\), or alternatively: \(A = P^{-1}LU\)

\[ \begin{align} Ax &= b \\ P^{-1}LUx &= b \\ \text{Let}\qquad Ux &= y \\ \text{Then solve}\qquad \left(P^{-1}L\right)y &= b \qquad \text{for}\,\, y \\ \text{Finally, solve}\qquad Ux &= y \qquad \text{for}\,\, x \end{align} \]

MATLAB Python
b = [9.1, 2, 0]';
[L,U,P] = lu(A);
y = mldivide(inv(P)*L, b);
x = mldivide(U, y);
>> x
x =
    5.0481
    4.0308
  -11.0923

% Verify the solution:
>> A*x - b
ans =
   1.0e-14 *

    0.1776
         0
         0

The small values of the solution error indicate that our LU solution is accurate. You should compare the LU solution with solution from mldivide(A, b).

LU, P = lu_factor(A)
b = np.array([9.1, 2, 0])
x = lu_solve((LU,P), b)

>>> x
array([  5.04807692,   4.03076923, -11.09230769])

# Verify the solution:
>>> np.dot(A, x) - b
array([  1.77635684e-15,   0.00000000e+00,   0.00000000e+00])

Matrix determinant

One way to test if a matrix is invertable is to calculate the determinant. If it is non-zero, then \(A\) is invertable.

Let \(A = \left(\begin{array}{cccc} 4 & 0 & 1 \\ 0 & 6 & 2 \\ 3.2 & 7 & 4 \end{array} \right)\)

MATLAB Python
A = [4, 0, 1; 0, 6, 2; 3.2, 7, 4];
>> det(A)
ans =
   20.8000
import numpy as np
from numpy.linalg import *
from scipy.linalg import *

A = np.array([[4, 0, 1],[0, 6, 2],[3.2, 7, 4]])
>>> det(A)
20.800000000000011

Matrix inverse

The matrix inverse is defined so that \(A A^{-1} = I\). Let \(A = \left(\begin{array}{cccc} 4 & 0 & 1 \\ 0 & 6 & 2 \\ 3.2 & 7 & 4 \end{array} \right)\)

MATLAB Python
A = [4, 0, 1; 0, 6, 2; 3.2, 7, 4];
>> inv(A)
ans =
    0.4808    0.3365   -0.2885
    0.3077    0.6154   -0.3846
   -0.9231   -1.3462    1.1538

>> A * inv(A)
ans =
    1.0000         0         0
   -0.0000    1.0000    0.0000
         0         0    1.0000
import numpy as np
from numpy.linalg import *
from scipy.linalg import *

A = np.array([[4, 0, 1],[0, 6, 2],[3.2, 7, 4]])
>>> inv(A)
array([[ 0.48076923,  0.33653846, -0.28846154],
       [ 0.30769231,  0.61538462, -0.38461538],
       [-0.92307692, -1.34615385,  1.15384615]])

>>> np.dot(A, inv(A))
array([[  1.00000000e+00,   0.00000000e+00,   0.00000000e+00],
       [ -2.22044605e-16,   1.00000000e+00,   4.44089210e-16],
       [  0.00000000e+00,   0.00000000e+00,   1.00000000e+00]])

Notice the small, essentially zero elements in the off-diagonal elements.

Matrix and vector norms

MATLAB Python
A = [4, 0, 1; 0, 6, 2; 3.2, 7, 4];
format long


% The default norm in MATLAB = max(svd(A))
>> norm(A)
ans =
  10.627892098461123
>> max(svd(A))
ans =
  10.627892098461123

% Other norms available: 
>> norm(A, 'fro')  % Frobenius norm: square root of
                   % the sum of squares of all entries in A
ans =
  11.499565209172042

% Other norms
% norm(A, 1)   : column sum norm: 13.0
% norm(A, inf) : row sum norm: 14.199999999999999
import numpy as np
from numpy.linalg import *
from scipy.linalg import *

A = np.array([[4, 0, 1],[0, 6, 2],[3.2, 7, 4]])

# The default norm in Python = Frobenius norm
>>> norm(A)
ans =
  11.499565209172042
>>> norm(A, 'fro')
ans =
  11.499565209172042

# Other norms available: 
>>> norm(A, 2)  % 2-norm: maximum singular value
ans =
  10.627892098461123

# Other norms
# norm(A, 1)      : column sum norm: 13.0
# norm(A, np.inf) : row sum norm: 14.199999999999999


Matrix conditioning

The condition number was defined as \({\rm cond}(A) \stackrel{\Delta}{=} \|A\|\times\|A^{-1}\|\). Any norm may be used, but if left unstated, then you should assume the 2-norm was used (maximum singular value).

MATLAB Python
A = [4, 0, 1; 0, 6, 2; 3.2, 7, 4];
format long

>> cond(A)
ans =
  23.724794742061690



% Condition numbers, calculated using other norms are also available

>> help cond
 COND   Condition number with respect to inversion.
    COND(X) returns the 2-norm condition number (the ratio of the
    largest singular value of X to the smallest).  Large condition
    numbers indicate a nearly singular matrix.
 
    COND(X,P) returns the condition number of X in P-norm:
 
       NORM(X,P) * NORM(INV(X),P). 
 
    where P = 1, 2, inf, or 'fro'.
import numpy as np
from numpy.linalg import *
from scipy.linalg import *

A = np.array([[4, 0, 1],[0, 6, 2],[3.2, 7, 4]])

>>> cond(A)
23.72479474206169

# Condition numbers, calculated using other norms are also available

>>> help(cond)
cond(x, p=None)
    Compute the condition number of a matrix.
    
    This function is capable of returning the condition number using
    one of seven different norms, depending on the value of `p` (see
    Parameters below).
    
    Parameters
    ----------
    x : array_like, shape (M, N)
        The matrix whose condition number is sought.
    p : {None, 1, -1, 2, -2, inf, -inf, 'fro'}, optional
        Order of the norm:
    
        =====  ============================
        p      norm for matrices
        =====  ============================
        None   2-norm, computed directly using the ``SVD``
        'fro'  Frobenius norm
        inf    max(sum(abs(x), axis=1))
        -inf   min(sum(abs(x), axis=1))
        1      max(sum(abs(x), axis=0))
        -1     min(sum(abs(x), axis=0))
        2      2-norm (largest sing. value)
        -2     smallest singular value
        =====  ============================
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