Eigenvalues and eigenvectors (eig) in SciPy - Step-by-Step Execution

Concept Flow - Eigenvalues and eigenvectors (eig)

Start with matrix A

↓

Compute eigenvalues and eigenvectors

↓

Eigenvalues (λ) found

↓

Eigenvectors (v) found

↓

Use λ and v for analysis or transformation

We start with a matrix, then calculate its eigenvalues and eigenvectors, which help us understand the matrix's key properties.

Execution Sample

SciPy

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

This code calculates eigenvalues and eigenvectors of matrix A using scipy.linalg.eig.

Execution Table

Step	Action	Matrix A	Eigenvalues (λ)	Eigenvectors (v)
1	Input matrix A	[[4, 2], [1, 3]]
2	Calculate characteristic polynomial	det(A - λI) = 0
3	Solve polynomial for λ		[5.0, 2.0]
4	Find eigenvectors for λ=5.0			[[0.8944, 0.7071], [0.4472, -0.7071]]
5	Find eigenvectors for λ=2.0			[[0.8944, 0.7071], [0.4472, -0.7071]]
6	Output eigenvalues and eigenvectors	[[4, 2], [1, 3]]	[5.0, 2.0]	[[0.8944, 0.7071], [0.4472, -0.7071]]
7	End

💡 All eigenvalues and eigenvectors computed for matrix A.

Variable Tracker

Variable	Start	After Step 3	After Step 6
A	[[4, 2], [1, 3]]	[[4, 2], [1, 3]]	[[4, 2], [1, 3]]
values	None	[5.0, 2.0]	[5.0, 2.0]
vectors	None	None	[[0.8944, 0.7071], [0.4472, -0.7071]]

Key Moments - 3 Insights

Why do eigenvectors have two columns?

Each column corresponds to an eigenvector for one eigenvalue, as shown in execution_table rows 4 and 5.

Why are eigenvalues sometimes complex numbers?

Eigenvalues can be complex if the matrix has no real eigenvalues; here they are real (row 3), but complex values can appear for other matrices.

What does the eigenvector represent physically?

It shows a direction that the matrix transformation stretches or shrinks without changing direction, as seen in rows 4 and 5.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what are the eigenvalues of matrix A?

A [5.0, 2.0]

B [4, 3]

C [0.8944, 0.4472]

D [1, 2]

💡 Hint

Check the 'Eigenvalues (λ)' column at step 3 in the execution_table.

At which step do we first see the eigenvectors computed?

A Step 2

B Step 4

C Step 3

D Step 6

💡 Hint

Look at the 'Eigenvectors (v)' column in the execution_table to find when vectors appear.

If matrix A was changed to [[0, -1], [1, 0]], what would likely happen to eigenvalues?

A They would remain real numbers

B They would be zero

C They would become complex numbers

D They would be the same as original

💡 Hint

Recall from key_moments that some matrices have complex eigenvalues.

Concept Snapshot

Eigenvalues and eigenvectors:
- Given matrix A, solve det(A - λI) = 0 for eigenvalues λ
- For each λ, find vector v where A v = λ v
- Use scipy.linalg.eig(A) to get values and vectors
- Eigenvalues can be real or complex
- Eigenvectors show directions unchanged by A

Full Transcript

This visual execution traces how to compute eigenvalues and eigenvectors of a matrix using scipy.linalg.eig. We start with matrix A, then calculate its eigenvalues by solving the characteristic polynomial. Next, eigenvectors are found for each eigenvalue. The execution table shows each step, including matrix input, polynomial solving, and eigenvector calculation. Variable tracking shows how values and vectors change from start to finish. Key moments clarify common confusions about eigenvector columns and complex eigenvalues. The quiz tests understanding of eigenvalues, eigenvector steps, and effects of matrix changes. The snapshot summarizes the process and key points for quick reference.

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