np.linalg.eig() for eigenvalues in NumPy - Step-by-Step Execution

Concept Flow - np.linalg.eig() for eigenvalues

Start with square matrix A

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Call np.linalg.eig(A)

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Compute eigenvalues and eigenvectors

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Return eigenvalues array and eigenvectors matrix

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Use eigenvalues and eigenvectors for analysis

The function takes a square matrix, calculates its eigenvalues and eigenvectors, and returns them for further use.

Execution Sample

NumPy

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

This code calculates eigenvalues and eigenvectors of matrix A and prints them.

Execution Table

Step	Action	Matrix A	Eigenvalues	Eigenvectors
1	Input matrix A	[[4 2] [1 3]]	-	-
2	Call np.linalg.eig(A)	[[4 2] [1 3]]	Computed	Computed
3	Calculate eigenvalues	[[4 2] [1 3]]	[5.0, 2.0]	-
4	Calculate eigenvectors	[[4 2] [1 3]]	[5.0, 2.0]	[[0.89442719 -0.70710678] [0.4472136 0.70710678]]
5	Return results	[[4 2] [1 3]]	[5.0, 2.0]	[[0.89442719 -0.70710678] [0.4472136 0.70710678]]

💡 Eigenvalues and eigenvectors computed and returned for matrix A

Variable Tracker

Variable	Start	After np.linalg.eig() call	Final
A	[[4 2] [1 3]]	[[4 2] [1 3]]	[[4 2] [1 3]]
eigenvalues	-	[5.0, 2.0]	[5.0, 2.0]
eigenvectors	-	[[0.89442719 -0.70710678] [0.4472136 0.70710678]]	[[0.89442719 -0.70710678] [0.4472136 0.70710678]]

Key Moments - 3 Insights

Why are eigenvalues returned as an array of numbers?

Eigenvalues are the special numbers that tell how the matrix stretches space. The execution_table row 3 shows these values computed as [5.0, 2.0].

What does the eigenvectors matrix represent?

Each column in the eigenvectors matrix (see execution_table row 4) is a vector that points in a direction that the matrix scales by the corresponding eigenvalue.

Why must the input matrix be square?

Eigenvalues and eigenvectors are defined only for square matrices. The input matrix A in execution_table row 1 is 2x2, which is square.

Visual Quiz - 3 Questions

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

A [5.0, 2.0]

B [4, 3]

C [1, 2]

D [0.894, 0.447]

💡 Hint

Check the 'Eigenvalues' column in execution_table row 3.

At which step in the execution_table are eigenvectors first shown?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look for the first row where 'Eigenvectors' column has values.

If matrix A was not square, what would happen when calling np.linalg.eig()?

A It would return eigenvalues and eigenvectors anyway

B It would raise an error

C It would return zeros

D It would return the input matrix unchanged

💡 Hint

Recall key_moments about matrix shape requirements.

Concept Snapshot

np.linalg.eig(matrix)
- Input: square matrix
- Output: eigenvalues array, eigenvectors matrix
- Eigenvalues: special scalars
- Eigenvectors: directions scaled by eigenvalues
- Used in data science for PCA, stability analysis

Full Transcript

This visual trace shows how np.linalg.eig() works step-by-step. We start with a square matrix A. Calling np.linalg.eig(A) computes eigenvalues and eigenvectors. Eigenvalues are numbers that describe how the matrix stretches space, and eigenvectors are directions that stretch by those numbers. The execution table shows each step: input matrix, computation, and output. Variables track the matrix and results. Key moments clarify common confusions like why the matrix must be square and what eigenvectors mean. The quiz tests understanding by asking about eigenvalues, when eigenvectors appear, and error handling for non-square matrices.