# 9.2. Linear Algebra¶

## 9.2.1. Rationale¶

>>> import numpy as np


Linear Algebra:

• np.sign()

• np.abs()

• np.sqrt()

• np.power()

Logarithms:

• np.log()

• np.log10()

• np.exp()

## 9.2.3. Determinant of a square matrix¶

>>> a = np.array([[1, 2, 3],
...               [4, 5, 6],
...               [7, 8, 9]])
>>>
>>> np.linalg.det(a)
0.0

>>> a = np.array([[4, 2, 0],
...               [9, 3, 7],
...               [1, 2, 1]])
>>>
>>> np.linalg.det(a)
-48.00000000000003


## 9.2.4. 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:

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


Multidimensional example:

>>> a = np.arange(24).reshape((2,3,4))
>>> b = np.arange(4)
>>>
>>> np.inner(a, b)
array([[ 14,  38,  62],
[ 86, 110, 134]])


## 9.2.5. Outer product¶

• np.outer()

Compute the outer product of two vectors

>>> a = np.array([1, 2, 3])
>>> b = np.array([4, 5, 6])
>>>
>>> np.outer(a, b)
array([[ 4,  5,  6],
[ 8, 10, 12],
[12, 15, 18]])


An example using a "vector" of letters:

>>> a = np.array(['a', 'b', 'c'])
>>>
>>> np.outer(a, [1, 2, 3])
Traceback (most recent call last):
numpy.core._exceptions._UFuncNoLoopError: ufunc 'multiply' did not contain a loop with signature matching types (dtype('<U1'), dtype('int64')) -> None

>>> 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)


## 9.2.6. Cross product¶

• np.cross()

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

Vector cross-product:

>>> a = [1, 2, 3]
>>> b = [4, 5, 6]
>>>
>>> np.cross(a, b)
array([-3,  6, -3])


One vector with dimension 2:

>>> a = [1, 2]
>>> b = [4, 5, 6]
>>>
>>> np.cross(a, b)
array([12, -6, -3])


## 9.2.7. 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.

>>> a = np.array([[1, 2, 3],
...               [4, 5, 6],
...               [7, 8, 9]])
>>>
>>> vals, vecs = np.linalg.eig(a)
>>>
>>> vals
array([ 1.61168440e+01, -1.11684397e+00, -9.75918483e-16])
>>>
>>> vecs
array([[-0.23197069, -0.78583024,  0.40824829],
[-0.52532209, -0.08675134, -0.81649658],
[-0.8186735 ,  0.61232756,  0.40824829]])


## 9.2.8. Inverse of a square matrix¶

>>> a = np.array([[1, 2, 3],
...               [4, 5, 6],
...               [7, 8, 9]])
>>>
>>> np.linalg.inv(a)
Traceback (most recent call last):
numpy.linalg.LinAlgError: Singular matrix

>>> a = np.array([[4, 2, 0],
...               [9, 3, 7],
...               [1, 2, 1]])
>>>
>>> b = np.linalg.inv(a)
>>> b
array([[ 0.22916667,  0.04166667, -0.29166667],
[ 0.04166667, -0.08333333,  0.58333333],
[-0.3125    ,  0.125     ,  0.125     ]])
>>>
>>> np.dot(a, b)
array([[1.00000000e+00, 5.55111512e-17, 0.00000000e+00],
[0.00000000e+00, 1.00000000e+00, 2.22044605e-16],
[0.00000000e+00, 1.38777878e-17, 1.00000000e+00]])


## 9.2.9. Singular value decomposition of a matrix¶

>>> a = np.array([[1, 2, 3],
...               [4, 5, 6],
...               [7, 8, 9]])
>>>
>>> U, s, Vh = np.linalg.svd(a)
>>>
>>> U
array([[-0.21483724,  0.88723069,  0.40824829],
[-0.52058739,  0.24964395, -0.81649658],
[-0.82633754, -0.38794278,  0.40824829]])
>>>
>>> s
array([1.68481034e+01, 1.06836951e+00, 3.33475287e-16])
>>>
>>> Vh
array([[-0.47967118, -0.57236779, -0.66506441],
[-0.77669099, -0.07568647,  0.62531805],
[-0.40824829,  0.81649658, -0.40824829]])


## 9.2.10. Linear Algebra¶

Table 9.2. 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 9.3. 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 9.4. Tensor operations

Function

Description

tensorsolve

Solve a linear tensor equation

tensorinv

Calculate an inverse of a tensor

Table 9.5. Exceptions

Function

Description

LinAlgError

Indicates a failed linear algebra operation

## 9.2.11. Assignments¶

• $$distance(a, b) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

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

"""
* 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

>>> assert euclidean_distance((0,0), (0,0)) is not Ellipsis, \
'Assign result to function: euclidean_distance'

>>> 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

# callable: Calculate distance between points using Euclidean algorithm
def euclidean_distance(a, b):
...


"""

* 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

>>> assert euclidean_distance((0,0), (0,0)) is not Ellipsis, \
'Assign result to function: euclidean_distance'

>>> 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

# callable: Calculate distance between points using Euclidean algorithm
def euclidean_distance(a, b):
pass