Eigenvalues and eigenvectors (eig) in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is an eigenvalue in the context of a matrix?

An eigenvalue is a special number that shows how much a matrix stretches or shrinks a vector when multiplied by it. It tells us the factor by which the vector changes length.

beginner

What is an eigenvector?

An eigenvector is a vector that does not change direction when a matrix multiplies it. It may get longer or shorter, but it points the same way.

beginner

How do you compute eigenvalues and eigenvectors using scipy?

You use the function scipy.linalg.eig(matrix). It returns two arrays: one for eigenvalues and one for eigenvectors.

intermediate

Why are eigenvalues and eigenvectors important in data science?

They help us understand data patterns, reduce data size, and find directions of maximum variance, like in Principal Component Analysis (PCA).

intermediate

What does it mean if an eigenvalue is zero?

A zero eigenvalue means the matrix squashes some vectors completely to zero, showing that the matrix is not invertible and loses some information.

Which function in scipy computes eigenvalues and eigenvectors?

A scipy.linalg.solve

B scipy.linalg.inv

C scipy.linalg.det

D scipy.linalg.eig

scipy.linalg.eig is the function used to compute eigenvalues and eigenvectors.

What does an eigenvector do when multiplied by its matrix?

A Changes length but not direction

B Remains unchanged

C Changes both length and direction

D Changes direction but not length

An eigenvector keeps its direction but its length can change when multiplied by the matrix.

If a matrix has an eigenvalue of zero, what does it imply?

A The matrix is diagonal

B The matrix is invertible

C The matrix is singular (not invertible)

D The matrix is identity

A zero eigenvalue means the matrix is singular and cannot be inverted.

Which of these is a use of eigenvalues and eigenvectors in data science?

A Sorting data alphabetically

B Reducing data dimensions with PCA

C Calculating averages

D Counting missing values

Eigenvalues and eigenvectors are used in PCA to reduce data dimensions.

What do the outputs of scipy.linalg.eig represent?

A Eigenvalues and eigenvectors

B Matrix inverse and determinant

C Matrix rank and trace

D Singular values and vectors

scipy.linalg.eig returns eigenvalues and eigenvectors of a matrix.

Explain in your own words what eigenvalues and eigenvectors are and why they matter.

Think about how a matrix changes vectors and what stays the same.

Describe how you would use scipy to find eigenvalues and eigenvectors of a matrix.

Focus on the function name and what it returns.

Practice

1. What do eigenvalues represent in the context of a matrix transformation?

easy

A. The sum of all matrix elements

B. The directions that change completely under the matrix

C. How much eigenvectors stretch or shrink when the matrix is applied

D. The inverse of the matrix

Solution

1. Step 1: Understand eigenvalues meaning

   Eigenvalues measure the factor by which eigenvectors are stretched or shrunk when multiplied by the matrix.

2. Step 2: Compare options with definition

   How much eigenvectors stretch or shrink when the matrix is applied correctly describes this stretching/shrinking effect. Other options describe unrelated concepts.

3. Final Answer:

   How much eigenvectors stretch or shrink when the matrix is applied -> Option C

4. Quick Check:

   Eigenvalues = stretch/shrink factor [OK]

Hint: Eigenvalues tell how vectors scale under the matrix [OK]

Common Mistakes:

• Confusing eigenvalues with eigenvectors
• Thinking eigenvalues are directions
• Mixing eigenvalues with matrix sum

2. Which of the following is the correct way to import the function to compute eigenvalues and eigenvectors using scipy?

easy

A. from scipy.linalg import eig

B. import scipy.eigen as eig

C. from scipy.linalg import eigenvalues

D. import scipy.linalg.eig as eigen

Solution

1. Step 1: Recall scipy function for eigenvalues

   The function to compute eigenvalues and eigenvectors is named 'eig' and is in scipy.linalg.

2. Step 2: Check import syntax

   from scipy.linalg import eig correctly imports 'eig' from 'scipy.linalg'. Other options use incorrect module names or syntax.

3. Final Answer:

   from scipy.linalg import eig -> Option A

4. Quick Check:

   Correct import = from scipy.linalg import eig [OK]

Hint: Use 'from scipy.linalg import eig' to get eigenvalues [OK]

Common Mistakes:

• Using wrong module names like scipy.eigen
• Trying to import non-existent functions
• Incorrect import syntax

3. What is the output of the following code?

import numpy as np
from scipy.linalg import eig
A = np.array([[2, 0], [0, 3]])
values, vectors = eig(A)
print(values)

medium

A. [2.+0.j 3.+0.j]

B. [0.+2.j 0.+3.j]

C. [5.+0.j 0.+0.j]

D. [2 3]

Solution

1. Step 1: Understand matrix and eig function

   Matrix A is diagonal with values 2 and 3 on the diagonal. Eigenvalues of a diagonal matrix are the diagonal elements.

2. Step 2: Predict output of eig

   scipy.linalg.eig returns eigenvalues as complex numbers, so output will be [2.+0.j 3.+0.j].

3. Final Answer:

   [2.+0.j 3.+0.j] -> Option A

4. Quick Check:

   Diagonal matrix eigenvalues = diagonal elements [OK]

Hint: Diagonal matrix eigenvalues = diagonal entries as complex numbers [OK]

Common Mistakes:

• Expecting real numbers instead of complex
• Adding eigenvalues instead of listing
• Confusing eigenvalues with eigenvectors

4. The following code throws an error. What is the main issue?

import numpy as np
from scipy.linalg import eig
A = np.array([[1, 2, 3], [4, 5, 6]])
values, vectors = eig(A)

medium

A. eig function is misspelled

B. Matrix A is not square

C. Missing import for numpy

D. Vectors variable name is invalid

Solution

1. Step 1: Check matrix shape

   Matrix A has shape (2, 3), which is not square (rows != columns).

2. Step 2: Understand eig requirements

   scipy.linalg.eig requires a square matrix to compute eigenvalues and eigenvectors.

3. Final Answer:

   Matrix A is not square -> Option B

4. Quick Check:

   eig needs square matrix [OK]

Hint: Check matrix shape; eig needs square matrix [OK]

Common Mistakes:

• Ignoring matrix shape requirement
• Assuming eig works on any matrix
• Blaming variable names or imports

5. Given the matrix B = [[4, 1], [2, 3]], which of the following is a correct eigenvalue computed using scipy.linalg.eig?

hard

A. 7.0 + 0.j

B. 3.0 + 0.j

C. 1.0 + 0.j

D. 5.0 + 0.j

Solution

1. Step 1: Calculate eigenvalues of B

   Eigenvalues satisfy det(B - λI) = 0. For B=[[4,1],[2,3]], characteristic polynomial is (4-λ)(3-λ)-2=0 -> λ² -7λ +10=0.

2. Step 2: Solve quadratic equation

   Roots are λ = (7 ± sqrt(49 - 40))/2 = (7 ± 3)/2 -> λ=5 or λ=2.

3. Step 3: Match options

   5.0 + 0.j is 5.0 + 0.j, which matches one eigenvalue. Others do not match.

4. Final Answer:

   5.0 + 0.j -> Option D

5. Quick Check:

   Eigenvalues = 5 and 2 [OK]

Hint: Solve characteristic polynomial for eigenvalues [OK]

Common Mistakes:

• Miscomputing characteristic polynomial
• Ignoring complex number format
• Choosing wrong root from quadratic