np.linalg.eig() for eigenvalues in NumPy

Introduction

We use np.linalg.eig() to find special numbers called eigenvalues and vectors from a square matrix. These help us understand important properties of the matrix.

When you want to find the main directions of data in machine learning.

To analyze stability in systems like weather or economics.

When solving physics problems involving vibrations or rotations.

To reduce data dimensions while keeping important information.

When studying graphs or networks to find key nodes.

Syntax

NumPy

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

matrix must be a square 2D numpy array.

The function returns two arrays: eigenvalues (1D) and eigenvectors (2D).

Examples

Find eigenvalues and eigenvectors of a simple diagonal matrix.

NumPy

import numpy as np
A = np.array([[2, 0], [0, 3]])
vals, vecs = np.linalg.eig(A)

Calculate eigenvalues and eigenvectors for a non-diagonal matrix.

NumPy

B = np.array([[4, 1], [2, 3]])
vals, vecs = np.linalg.eig(B)

Example with symmetric matrix to see real eigenvalues.

NumPy

C = np.array([[1, 2], [2, 1]])
vals, vecs = np.linalg.eig(C)

Sample Program

This program finds eigenvalues and eigenvectors of a 2x2 matrix. It prints both so you can see the special numbers and directions.

NumPy

import numpy as np

# Define a 2x2 matrix
matrix = np.array([[5, 4], [1, 2]])

# Calculate eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eig(matrix)

print("Eigenvalues:")
print(eigenvalues)
print("\nEigenvectors:")
print(eigenvectors)

Important Notes

Eigenvalues can be complex numbers if the matrix is not symmetric.

Eigenvectors are normalized to length 1 by default.

Order of eigenvalues matches the columns of eigenvectors.

Summary

np.linalg.eig() finds eigenvalues and eigenvectors of square matrices.

Eigenvalues tell you about scaling factors; eigenvectors show directions.

This is useful in many fields like data science, physics, and engineering.

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