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Here, we will aim to provide you with a number of exercises and solutions with a varying degree of difficulty. There will also occur exercises that requires you to use functions that has not yet been described in Matrices and linear algebra in Python.

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Determinant

Exercise 1: Simple matrix

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title	Exercise: Simple matrix

Compute the determinant

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title	Solution: Simple matrix

Code Block

language	py
title	Solution: Simple matrix

import numpy as np 

# Defining the matrix
A = np.array([[1, 3, 2],
            [4, 1, 3],
            [2, 5, 2]])

det = np.linalg.det(A)
print(det)


# Output:
16.999999999999993   # e.i. det(A) = 17

Exercise 2: Complex numbers

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title	Exercise: Complex numbers

This example is taken from an example exam in TMA4110 - Calculus 3, check out the exam and the solution for hand calculations (only in norwegainNorwegian). This is task 4.

Comput the determinant

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title	Solution: Complex numbers

Code Block

language	py
title	Solution: Complex Numbers

import numpy as np 

# Defining the matrix
# Notice how we can define complex numbers in python by adding the letter "j"
# "j" is used to denote an imaginary unit instead of "i", because Python follows engineering,
# where "i" denotes the electric current as a function of time (especially electrial engineering)
A = np.array([[3, 1-1j, 1j, 4],
            [3, 1, 1-2j, 4+7j],
            [6j, 2+2j, -2, 3j],
            [-3, -1+1j, 1, 3-4j]], dtype=complex)

det = np.linalg.det(A)
print(det)


# Output:
(-15-15j)   # e.i. det(A) = -15-15i

Eigenvalues and eigenvectors

Exercise 1

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title	Exercise 1

Find the eigenvalues and corresponding eigenvectors to the following matrix :A, where

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title	Solution

Code Block

language	py
title	Solution

import numpy as np 

A = np.array([[1, 2], 
            [2, 1]])
 
w, v = np.linalg.eig(A)     # "w" are the eigenvalues, "v" the eigenvectors
 
print(w)
print(v)


# Output:
[ 3. -1.]
[[ 0.70710678 -0.70710678]
 [ 0.70710678  0.70710678]]

We can interpret this as

and

.

Exercise 2

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title	Exercise 2

Find the eigenvalues and corresponding eigenvectors to

the following

matrix

:

A, where

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title	Solution

Code Block

language	py
title	Solution

import numpy as np 

A = np.array([[0, 1], 
            [0, 0]])
 
w, v = np.linalg.eig(A)     # "w" are the eigenvalues, "v" the eigenvectors
 
print(w)
print(v)


# Output:
[0. 0.]
[[ 1.00000000e+000 -1.00000000e+000]
 [ 0.00000000e+000  2.00416836e-292]]	# ...e-292 ~ 0

We can interpret this as

, which is one eigenvalue and one eigenvector (since

, thus linearly dependent). If this is the case, why does np.linalg.eig then return the same eigenvalue twice with two different eigenvectors?

Simplified, np.linalg.eig of an

matrix is guaranteed to always return

eigenvalues and eigenvectors. In the function documentation it says: "If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent." In this example we know that two eigenvectors are going to be produced. If they're actually the same vector, this will be indicated by making the eigenvalues identical.

Note
This does not imply that all eigenvectors with the same eigenvalue are linearly dependant! To further clarify this, take a look at exercise 3.

Exercise 3

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title	Exercise 3

Find the eigenvalues and corresponding eigenvectors to the following matrix :A, where

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title	Solution

Code Block

language	py
title	Solution

import numpy as np 

A = np.array([[4, 2, 3], 
            [-1, 1, -3],
            [2, 4, 9]])
 
w, v = np.linalg.eig(A)     # "w" are the eigenvalues, "v" the eigenvectors
 
print(w)
print(v)


# Output:
[8. 3. 3.]
[[-0.40824829  0.0153225   0.9635814 ]
 [ 0.40824829 -0.83430241 -0.14040935]
 [-0.81649658  0.55109411 -0.22758756]]

As in exercise 2, the same eigenvalue are produced several times, though it's not the same eigenvectors this time. We can interpret the output as

and

, where

and

are linearly independent and together spans a plane. The same plane can be span by the e.g. the vectors

and

for easier interpretation, though both solution are correct.

Inverse

Exercise 1

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title	Exercise

Find the inverse of the matrix

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title	Solution

Code Block

language	py
title	Solution

import numpy as np 

A = np.array([[2, 2, 0],
            [0, 0, 1],
            [4, 2, 0]])
inv_A = np.linalg.inv(A)
print(inv_A)


# Output:
[[-0.5  0.   0.5]
 [ 1.   0.  -0.5]
 [ 0.   1.   0. ]]

Least squares solution to a linear system

Exercise 1

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title	Exercise

This example is taken from an example exam in TMA4110 - Calculus 3, check out the exam and the solution for hand calculations (only in norwegainNorwegian). This is task 5.

Use the least squares method to find an approximation to the linear system

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title	Solution

First, lets write the equations as matrices on the format

, where

To solve the problem in NumPy, we will use the function numpy.linalg.lstsq (or with SciPys scipy.linalg.lstsq) which is currently not described in Matrices and linear algebra in Python. We recommend you to read the function documentation before proceeding.

Code Block

language	py
title	Solution

import numpy as np 

# Defining the matrices
A = np.array([[1, 2],
            [3, 4],
            [5, 6]])
b = np.array([[1],[2],[3]])

x, residuals, rank, s = np.linalg.lstsq(A, b, rcond=None)   
# "rcond=None" sets new default for rcond, see function documentation

print(x)    # "x" is the solution


# Output:
[[-5.97106181e-17]
 [ 5.00000000e-01]]

We can interpret this such as the least squares solution is

Solving a system of linear equations

Exercise 1

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title	Exercise 1

Solve the following system of linear equations:

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title	Solution

Code Block

language	py
title	Solution

import numpy as np 

A = np.array([[2, -4, 9],       # Right-hand side
            [4, -3, 8],
            [-2, 4, -2]])
b = np.array([-38, -26, 17])    # Left-hand side

solution = np.linalg.solve(A, b)
print(solution)


# Output:
[ 2.5  4.  -3. ]

This is interpreted as

and

Exercise 2: Singular matrix

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title	Exercise 2

Solve the following system of linear equations:

Expand

title	Solution

Code Block

language	py
title	Solution

import numpy as np 

A = np.array([[1, 3, 6],       	# Right-hand side
            [2, 8, 16],
            [2, 6, 12]])
b = np.array([4, 8, 8])     	# Left-hand side

solution = np.linalg.solve(A, b)

print(solution)


# Output:
numpy.linalg.LinAlgError: Singular matrix

As we can see, using np.linalg.solve returns a LinAlgError telling us that the matrix (in this case A) is a singular matrix. A singular matrix is one that is not invertible. This means that the system of equations we are trying to solve does not have a unique solution (either none or multiple); np.linalg.solve can't handle this.

By using np.linalg.lstsq (least squares solution) instead, we will at least get one solution.

Code Block

language	py
title	Solution using least squares method

import numpy as np 

A = np.array([[1, 3, 6],       	# Right-hand side
            [2, 8, 16],
            [2, 6, 12]])
b = np.array([4, 8, 8])     	# Left-hand side

x, residuals, rank, s = np.linalg.lstsq(A, b, rcond=None) 
# "rcond=None" sets new default for rcond, see function documentation

print(x)    # "x" is the solution


# Output:
[4. 0. 0.]

This is interpreted as

and

Note
Keep in mind that this is only one of multiple solutions!

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Key

Determinant

Exercise 1: Simple matrix

Exercise 2: Complex numbers

Eigenvalues and eigenvectors

Exercise 1

Exercise 2

Exercise 3

Inverse

Exercise 1

Least squares solution to a linear system

Exercise 1

Solving a system of linear equations

Exercise 1

Exercise 2: Singular matrix