5.4. Linear Algebra¶

5.4.1. Linear Algebra¶

np.sign()
np.abs()
np.sqrt()
np.power()

5.4.2. Logarithms¶

np.log()
np.log10()
np.exp()

5.4.3. Vector and matrix mathematics¶

5.4.4. Determinant of a square matrix¶

import numpy as np


a = np.array([[4, 2, 0], [9, 3, 7], [1, 2, 1]], float)
# array([[ 4., 2., 0.],
#        [ 9., 3., 7.],
#        [ 1., 2., 1.]])

np.linalg.det(a)
# -53.999999999999993

5.4.5. Inner product¶

Compute inner product of two vectors
np.inner()
Ordinary inner product of vectors for 1-D arrays (without complex conjugation)
In higher dimensions a sum product over the last axes

Ordinary inner product for vectors:

import numpy as np


a = np.array([1, 2, 3])
b = np.array([0, 1, 0])

np.inner(a, b)
# 2

Multidimensional example:

import numpy as np


a = np.arange(24).reshape((2,3,4))
b = np.arange(4)

np.inner(a, b)
# array([[ 14,  38,  62],
#        [ 86, 110, 134]])

5.4.6. Outer product¶

Compute the outer product of two vectors
np.outer()

import numpy as np


a = np.array([1, 4, 0], float)
b = np.array([2, 2, 1], float)

np.outer(a, b)
# array([[ 2., 2., 1.],
#        [ 8., 8., 4.],
#        [ 0., 0., 0.]])

An example using a "vector" of letters:

import numpy as np


a = np.array(['a', 'b', 'c'], dtype=object)

np.outer(a, [1, 2, 3])
# array([['a', 'aa', 'aaa'],
#        ['b', 'bb', 'bbb'],
#        ['c', 'cc', 'ccc']], dtype=object)

5.4.7. Cross product¶

The cross product of a and b in R^3 is a vector perpendicular to both a and b
np.cross()

Vector cross-product:

import numpy as np


x = [1, 2, 3]
y = [4, 5, 6]

np.cross(x, y)
# array([-3,  6, -3])

One vector with dimension 2:

import numpy as np


x = [1, 2]
y = [4, 5, 6]

np.cross(x, y)
# array([12, -6, -3])

5.4.8. Eigenvalues and vectors of a square matrix¶

Each of a set of values of a parameter for which a differential equation has a nonzero solution (an eigenfunction) under given conditions
Any number such that a given matrix minus that number times the identity matrix has a zero determinant

import numpy as np


a = np.array([[4, 2, 0], [9, 3, 7], [1, 2, 1]], float)
# array([[ 4., 2., 0.],
#        [ 9., 3., 7.],
#        [ 1., 2., 1.]])

vals, vecs = np.linalg.eig(a)

vals
# array([ 9. , 2.44948974, -2.44948974])

vecs
# array([[-0.3538921 , -0.56786837, 0.27843404],
#        [-0.88473024, 0.44024287, -0.89787873],
#        [-0.30333608, 0.69549388, 0.34101066]])

5.4.9. Inverse of a square matrix¶

import numpy as np


a = np.array([[4, 2, 0], [9, 3, 7], [1, 2, 1]], float)
# array([[ 4., 2., 0.],
#        [ 9., 3., 7.],
#        [ 1., 2., 1.]])

np.linalg.inv(a)
# array([[ 0.14814815, 0.07407407, -0.25925926],
#        [ 0.2037037 , -0.14814815, 0.51851852],
#        [-0.27777778, 0.11111111, 0.11111111]])

import numpy as np


a = np.array([[4, 2, 0], [9, 3, 7], [1, 2, 1]], float)
b = np.linalg.inv(a)

np.dot(a, b)
# array([[ 1.00000000e+00, 5.55111512e-17, 2.22044605e-16],
#        [ 0.00000000e+00, 1.00000000e+00, 5.55111512e-16],
#        [ 1.11022302e-16, 0.00000000e+00, 1.00000000e+00]])

5.4.10. Singular value decomposition of a matrix¶

import numpy as np


a = np.array([[1, 3, 4], [5, 2, 3]], float)

U, s, Vh = np.linalg.svd(a)

U
# array([[-0.6113829 , -0.79133492],
#        [-0.79133492, 0.6113829 ]])

s
# array([ 7.46791327, 2.86884495])

Vh
# array([[-0.61169129, -0.45753324, -0.64536587],
#        [ 0.78971838, -0.40129005, -0.46401635],
#        [-0.046676 , -0.79349205, 0.60678804]])

5.4.11. Linear Algebra¶

Table 5.2. Linear algebra basics¶
Function	Description
norm	Vector or matrix norm
inv	Inverse of a square matrix
solve	Solve a linear system of equations
det	Determinant of a square matrix
slogdet	Logarithm of the determinant of a square matrix
lstsq	Solve linear least-squares problem
pinv	Pseudo-inverse (Moore-Penrose) calculated using a singular value decomposition
matrix_power	Integer power of a square matrix
matrix_rank	Calculate matrix rank using an SVD-based method

Table 5.3. Eigenvalues and decompositions¶
Function	Description
eig	Eigenvalues and vectors of a square matrix
eigh	Eigenvalues and eigenvectors of a Hermitian matrix
eigvals	Eigenvalues of a square matrix
eigvalsh	Eigenvalues of a Hermitian matrix
qr	QR decomposition of a matrix
svd	Singular value decomposition of a matrix
cholesky	Cholesky decomposition of a matrix

Table 5.4. Tensor operations¶
Function	Description
tensorsolve	Solve a linear tensor equation
tensorinv	Calculate an inverse of a tensor

Table 5.5. Exceptions¶
Function	Description
LinAlgError	Indicates a failed linear algebra operation

5.4.12. Assignments¶

../_images/euclidean-distance.png — Figure 5.11. Calculate Euclidean distance in Cartesian coordinate system¶

\(distance(a, b) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
\(distance(a, b) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + ... + (n_2 - n_1)^2}\)

Code 5.64. Solution¶

"""
* Assignment: Numpy Algebra Euclidean 2D
* Complexity: easy
* Lines of code: 6 lines
* Time: 5 min

English:
    1. Given are two points `a: tuple[int, int]` and `b: tuple[int, int]`
    2. Coordinates are in cartesian system
    3. Points `a` and `b` are in two dimensional space
    4. Calculate distance between points using Euclidean algorithm
    5. Run doctests - all must succeed

Polish:
    1. Dane są dwa punkty `a: tuple[int, int]` i `b: tuple[int, int]`
    2. Koordynaty są w systemie kartezjańskim
    3. Punkty `a` i `b` są w dwuwymiarowej przestrzeni
    4. Oblicz odległość między nimi wykorzystując algorytm Euklidesa
    5. Uruchom doctesty - wszystkie muszą się powieść

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> a = (1, 0)
    >>> b = (0, 1)
    >>> euclidean_distance(a, b)
    1.4142135623730951

    >>> euclidean_distance((0,0), (1,0))
    1.0

    >>> euclidean_distance((0,0), (1,1))
    1.4142135623730951

    >>> euclidean_distance((0,1), (1,1))
    1.0

    >>> euclidean_distance((0,10), (1,1))
    9.055385138137417
"""

from math import sqrt


def euclidean_distance(a, b):
    pass

Code 5.65. Solution¶

"""

* Assignment: Numpy Algebra Euclidean Ndim
* Complexity: easy
* Lines of code: 7 lines
* Time: 8 min

English:
    1. Given are two points `a: Sequence[int]` and `b: Sequence[int]`
    2. Coordinates are in cartesian system
    3. Points `a` and `b` are in n-dimensional space
    4. Points `a` and `b` must be in the same space
    5. Calculate distance between points using Euclidean algorithm
    6. Run doctests - all must succeed

Polish:
    1. Dane są dwa punkty `a: Sequence[int]` i `b: Sequence[int]`
    2. Koordynaty są w systemie kartezjańskim
    3. Punkty `a` i `b` są w n-wymiarowej przestrzeni
    4. Punkty `b` i `b` muszą być w tej samej przestrzeni
    5. Oblicz odległość między nimi wykorzystując algorytm Euklidesa
    6. Uruchom doctesty - wszystkie muszą się powieść

Hints:
    * `for n1,n2 in zip(a,b)`

Tests:
    >>> import sys; sys.tracebacklimit = 0

    >>> euclidean_distance((0,0,1,0,1), (1,1))
    Traceback (most recent call last):
    ValueError: Points must be in the same dimensions

    >>> euclidean_distance((0,0,0), (0,0,0))
    0.0

    >>> euclidean_distance((0,0,0), (1,1,1))
    1.7320508075688772

    >>> euclidean_distance((0,1,0,1), (1,1,0,0))
    1.4142135623730951

    >>> euclidean_distance((0,0,1,0,1), (1,1,0,0,1))
    1.7320508075688772
"""

from math import sqrt


def euclidean_distance(a, b):
    pass