np.linalg.eig() for eigenvalues in NumPy - Mini Project: Build & Apply

Finding Eigenvalues and Eigenvectors with np.linalg.eig()

📖 Scenario: Imagine you are analyzing a simple system where you want to understand its main directions of change. This is common in physics, engineering, and data science. Eigenvalues and eigenvectors help us find these directions.

🎯 Goal: You will create a matrix, use np.linalg.eig() to find its eigenvalues and eigenvectors, and then display the results.

📋 What You'll Learn

Create a 2x2 numpy array called matrix with values [[4, 2], [1, 3]]

Create a variable called eigenvalues and a variable called eigenvectors to store the results of np.linalg.eig(matrix)

Print the eigenvalues and eigenvectors variables

💡 Why This Matters

🌍 Real World

Eigenvalues and eigenvectors help in physics to understand vibrations, in data science for principal component analysis, and in engineering for system stability.

💼 Career

Many data science and engineering jobs require understanding matrix properties and using eigenvalues for dimensionality reduction and system analysis.

1

Create the matrix

Create a 2x2 numpy array called matrix with these exact values: [[4, 2], [1, 3]]

NumPy

import numpy as np
# Create the matrix variable here
# Your code here

Hint

Use np.array() to create the matrix with the given values.

2

Calculate eigenvalues and eigenvectors

Use np.linalg.eig() on the matrix to get eigenvalues and eigenvectors. Store them in variables called eigenvalues and eigenvectors respectively.

NumPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
# Calculate eigenvalues and eigenvectors here
# Your code here

Hint

Use eigenvalues, eigenvectors = np.linalg.eig(matrix) to get both results at once.

3

Print eigenvalues

Print the variable eigenvalues to display the eigenvalues of the matrix.

NumPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
eigenvalues, eigenvectors = np.linalg.eig(matrix)
# Print eigenvalues here
# Your code here

Hint

Use print(eigenvalues) to show the eigenvalues.

4

Print eigenvectors

Print the variable eigenvectors to display the eigenvectors of the matrix.

NumPy

import numpy as np
matrix = np.array([[4, 2], [1, 3]])
eigenvalues, eigenvectors = np.linalg.eig(matrix)
print(eigenvalues)
# Print eigenvectors here
# Your code here

Hint

Use print(eigenvectors) to show the eigenvectors.

Practice

1. What does the function np.linalg.eig() return when applied to a square matrix?

easy

A. The determinant and inverse of the matrix

B. An array of eigenvalues and a matrix of eigenvectors

C. The transpose and trace of the matrix

D. The sum and product of matrix elements

Solution

1. Step 1: Understand the purpose of np.linalg.eig()

   This function is designed to find eigenvalues and eigenvectors of a square matrix.

2. Step 2: Recall the output format

   It returns two objects: one array with eigenvalues and one matrix with eigenvectors as columns.

3. Final Answer:

   An array of eigenvalues and a matrix of eigenvectors -> Option B

4. Quick Check:

   Eigenvalues and eigenvectors [OK]

Hint: Remember: eig() returns eigenvalues and eigenvectors [OK]

Common Mistakes:

• Confusing eigenvalues with determinant
• Expecting only one output instead of two
• Mixing eigenvectors with matrix transpose

2. Which of the following is the correct syntax to compute eigenvalues and eigenvectors of matrix A using NumPy?

easy

A. eigenvalues, eigenvectors = np.linalg.eig(A)

B. eigenvalues = np.linalg.eigvals(A)

C. eigenvectors, eigenvalues = np.linalg.eig(A)

D. eigenvalues, eigenvectors = np.eig.linalg(A)

Solution

1. Step 1: Recall the correct function and output order

   The function np.linalg.eig() returns eigenvalues first, then eigenvectors.

2. Step 2: Check syntax correctness

   eigenvalues, eigenvectors = np.linalg.eig(A) correctly assigns eigenvalues and eigenvectors in order from np.linalg.eig(A).

3. Final Answer:

   eigenvalues, eigenvectors = np.linalg.eig(A) -> Option A

4. Quick Check:

   Correct function and order [OK]

Hint: eig() returns (values, vectors) in that order [OK]

Common Mistakes:

• Swapping eigenvalues and eigenvectors in assignment
• Using wrong function like np.linalg.eigvals() for both outputs
• Incorrect module or function name

3. Given the matrix A = np.array([[2, 0], [0, 3]]), what will be the output of np.linalg.eig(A)[0]?

medium

A. [3. 2.]

B. [0. 0.]

C. [2. 3.]

D. [5. 0.]

Solution

1. Step 1: Identify eigenvalues of a diagonal matrix

   For a diagonal matrix, eigenvalues are the diagonal elements: 2 and 3.

2. Step 2: Check the output of np.linalg.eig(A)[0]

   This returns the eigenvalues array, which will be [2. 3.].

3. Final Answer:

   [2. 3.] -> Option C

4. Quick Check:

   Diagonal elements = eigenvalues [OK]

Hint: Diagonal matrix eigenvalues = diagonal elements [OK]

Common Mistakes:

• Confusing eigenvalues order
• Expecting eigenvectors instead of eigenvalues
• Misreading matrix elements

4. What is wrong with this code snippet?

import numpy as np
A = np.array([[1, 2], [3, 4]])
eigenvalues, eigenvectors = np.linalg.eigvals(A)

medium

A. np.linalg.eigvals() returns only eigenvalues, not eigenvectors

B. Matrix A is not square

C. np.linalg.eigvals() requires two arguments

D. The import statement is incorrect

Solution

1. Step 1: Understand the function np.linalg.eigvals()

   This function returns only eigenvalues, not eigenvectors.

2. Step 2: Check the assignment in the code

   The code tries to unpack two values, but eigvals() returns only one, causing an error.

3. Final Answer:

   np.linalg.eigvals() returns only eigenvalues, not eigenvectors -> Option A

4. Quick Check:

   eigvals() returns one output [OK]

Hint: eigvals() returns only eigenvalues, not vectors [OK]

Common Mistakes:

• Expecting two outputs from eigvals()
• Thinking matrix must be non-square
• Misunderstanding import syntax

5. You have a matrix B = np.array([[1, 0], [0, -3]]). You want to find the eigenvalue with the largest magnitude. Which code snippet correctly finds it?

hard

A. vals, vecs = np.linalg.eig(B)
largest = max(vecs)

B. vals = np.linalg.eigvals(B)
largest = max(vals)

C. vals = np.linalg.eigvals(B)
largest = vals[np.argmax(vals)]

D. vals, vecs = np.linalg.eig(B)
largest = vals[np.argmax(np.abs(vals))]

Solution

1. Step 1: Compute eigenvalues and eigenvectors

   Use np.linalg.eig(B) to get both eigenvalues and eigenvectors.

2. Step 2: Find eigenvalue with largest magnitude

   Use np.abs(vals) to get absolute values, then np.argmax() to find index of largest magnitude eigenvalue.

3. Step 3: Select eigenvalue at that index

   Indexing vals with that index gives the eigenvalue with largest magnitude.

4. Final Answer:

   vals, vecs = np.linalg.eig(B) largest = vals[np.argmax(np.abs(vals))] -> Option D

5. Quick Check:

   Use abs and argmax on eigenvalues [OK]

Hint: Use np.abs() and np.argmax() on eigenvalues to find largest [OK]

Common Mistakes:

• Using max() directly without abs()
• Trying to find max of eigenvectors
• Using wrong function for eigenvalues