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

Concept Flow - Eigenvalues and eigenvectors (eig)

Start with matrix A

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Compute eigenvalues and eigenvectors

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Eigenvalues (λ) found

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Eigenvectors (v) found

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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.