Cholesky decomposition in SciPy - Interactive Code Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to import the Cholesky decomposition function from scipy.linalg.

SciPy

from scipy.linalg import [1]

# Now you can use the function for decomposition

Drag options to blanks, or click blank then click option'

Aeig

Binv

Ccholesky

Dsvd

Attempts:

3 left

💡 Hint

Common Mistakes

Importing the wrong function like 'inv' or 'eig'.

Using 'svd' which is for singular value decomposition, not Cholesky.

The cholesky function from scipy.linalg computes the Cholesky decomposition of a matrix.

2fill in blank

medium

Complete the code to perform Cholesky decomposition on matrix 'A'.

SciPy

import numpy as np
from scipy.linalg import cholesky

A = np.array([[4, 2], [2, 3]])
L = cholesky([1], lower=True)
print(L)

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AA

BL

Cnp

Dcholesky

Attempts:

3 left

💡 Hint

Common Mistakes

Passing the function name instead of the matrix.

Passing the result variable 'L' before it is assigned.

The matrix A is passed to the cholesky function to compute its Cholesky decomposition.

3fill in blank

hard

Fix the error in the code to correctly compute the Cholesky decomposition.

SciPy

import numpy as np
from scipy.linalg import cholesky

B = np.array([[1, 2], [2, 1]])
L = cholesky([1], lower=True)
print(L)

Drag options to blanks, or click blank then click option'

Anp

BB

Ccholesky

DL

Attempts:

3 left

💡 Hint

Common Mistakes

Passing the function name or other variables instead of the matrix.

Using a matrix that is not positive definite (this matrix is not positive definite, so decomposition will fail).

The matrix B is passed to the cholesky function. The matrix must be positive definite for Cholesky decomposition to work.

4fill in blank

hard

Fill both blanks to create a dictionary of Cholesky decompositions for matrices with keys 'X' and 'Y'.

SciPy

matrices = {'X': np.array([[9, 3], [3, 5]]), 'Y': np.array([[16, 4], [4, 10]])}
decompositions = {key: [1](mat, lower=True) for key, mat in matrices.items() if mat.shape[0] == [2]
print(decompositions)

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Acholesky

B2

C1

Dnp.linalg.cholesky

Attempts:

3 left

💡 Hint

Common Mistakes

Using the wrong function name like 'np.linalg.cholesky' when importing from scipy.linalg.

Checking for wrong matrix size.

The cholesky function is used to decompose each matrix. The condition checks if the matrix is 2x2 by verifying mat.shape[0] == 2.

5fill in blank

hard

Fill all three blanks to create a dictionary comprehension that stores the upper triangular Cholesky decomposition for each matrix in 'data'.

SciPy

data = {'A': np.array([[25, 15], [15, 18]]), 'B': np.array([[36, 12], [12, 20]])}
results = { [1]: cholesky(mat, lower=[2]) for [3], mat in data.items() }
print(results)

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A'A'

BFalse

Ckey

DTrue

Attempts:

3 left

💡 Hint

Common Mistakes

Using a string literal instead of the variable for keys.

Setting lower=True instead of False.

Not unpacking the dictionary items correctly.

The dictionary comprehension uses key as the dictionary key. Setting lower=False computes the upper triangular matrix. The loop unpacks key, mat from data.items().

Practice

1. What does Cholesky decomposition do to a matrix?

easy

A. It breaks the matrix into a lower-triangular matrix and its transpose.

B. It converts the matrix into a diagonal matrix only.

C. It finds the inverse of the matrix directly.

D. It changes the matrix into a symmetric matrix.

Solution

1. Step 1: Understand Cholesky decomposition

   Cholesky decomposition splits a symmetric, positive definite matrix into a product of a lower-triangular matrix and its transpose.

2. Step 2: Compare options with definition

   Only It breaks the matrix into a lower-triangular matrix and its transpose. correctly describes this decomposition as lower-triangular matrix times its transpose.

3. Final Answer:

   It breaks the matrix into a lower-triangular matrix and its transpose. -> Option A

4. Quick Check:

   Cholesky = Lower-triangular x Transpose [OK]

Hint: Cholesky = lower-triangular matrix x its transpose [OK]

Common Mistakes:

• Confusing Cholesky with diagonalization
• Thinking it finds matrix inverse
• Assuming it changes matrix symmetry

2. Which of the following is the correct way to import and use Cholesky decomposition from scipy?

easy

A. from scipy.linalg import cholesky
L = cholesky(A, lower=True)

B. import scipy.cholesky as chol
L = chol.decompose(A)

C. from scipy.linalg import cholesky
L = cholesky(A, upper=True)

D. import cholesky from scipy
L = cholesky(A)

Solution

1. Step 1: Recall correct import syntax

   The correct import is from scipy.linalg import cholesky.

2. Step 2: Check usage parameters

   To get the lower-triangular matrix, use cholesky(A, lower=True). from scipy.linalg import cholesky L = cholesky(A, lower=True) matches this exactly.

3. Final Answer:

   from scipy.linalg import cholesky\nL = cholesky(A, lower=True) -> Option A

4. Quick Check:

   Correct import and lower=True parameter [OK]

Hint: Use from scipy.linalg import cholesky and lower=True [OK]

Common Mistakes:

• Using wrong import path
• Confusing upper and lower parameters
• Incorrect function call syntax

3. Given the matrix A = [[4, 2], [2, 3]], what is the output of scipy.linalg.cholesky(A, lower=True)?

medium

A. [[4.0, 2.0], [2.0, 3.0]]

B. [[2.0, 1.0], [0.0, 1.73205081]]

C. [[2.0, 0.0], [1.0, 1.41421356]]

D. [[1.0, 0.0], [0.5, 1.0]]

Solution

1. Step 1: Understand Cholesky on matrix A

   Matrix A is symmetric and positive definite, so cholesky(A, lower=True) returns lower-triangular L such that A = L x L.T.

2. Step 2: Calculate L manually or verify

   L = [[2.0, 0.0], [1.0, 1.41421356]] satisfies A = L x L.T.

3. Final Answer:

   [[2.0, 0.0], [1.0, 1.41421356]] -> Option C

4. Quick Check:

   Lower-triangular matrix matches [[2.0, 0.0], [1.0, 1.41421356]] [OK]

Hint: Check lower-triangular matrix matches A = L x L.T [OK]

Common Mistakes:

• Confusing upper and lower matrices
• Using original matrix as output
• Incorrect decimal approximations

4. What is the error in this code snippet?

import numpy as np
from scipy.linalg import cholesky
A = np.array([[1, 2], [3, 4]])
L = cholesky(A, lower=True)

medium

A. Parameter 'lower' is invalid in cholesky.

B. Missing import for numpy.linalg.

C. Matrix A must be integer type.

D. Matrix A is not symmetric positive definite, so cholesky fails.

Solution

1. Step 1: Check matrix properties

   Matrix A = [[1,2],[3,4]] is not symmetric (A != A.T) and not positive definite.

2. Step 2: Understand cholesky requirements

   Cholesky decomposition requires symmetric positive definite matrices; otherwise, it raises an error.

3. Final Answer:

   Matrix A is not symmetric positive definite, so cholesky fails. -> Option D

4. Quick Check:

   Cholesky needs symmetric positive definite matrix [OK]

Hint: Check matrix symmetry and positive definiteness before cholesky [OK]

Common Mistakes:

• Ignoring matrix symmetry requirement
• Assuming any matrix works
• Misunderstanding parameter names

5. You have a covariance matrix C = [[2, -1], [-1, 2]]. How can you use Cholesky decomposition to generate correlated random samples X from independent standard normal samples Z?

hard

A. Use L = cholesky(C, lower=True) and compute X = Z @ L

B. Use L = cholesky(C, lower=True) and compute X = L @ Z

C. Use L = cholesky(C, upper=True) and compute X = L @ Z

D. Use L = cholesky(C, lower=True) and compute X = L.T @ Z

Solution

1. Step 1: Decompose covariance matrix

   Compute L = cholesky(C, lower=True) so that C = L x L.T.

2. Step 2: Generate correlated samples

   Multiply L by independent samples Z: X = L @ Z to introduce correlation matching C.

3. Final Answer:

   Use L = cholesky(C, lower=True) and compute X = L @ Z -> Option B

4. Quick Check:

   Correlated samples = L x independent samples [OK]

Hint: Multiply lower-triangular L by independent samples Z [OK]

Common Mistakes:

• Multiplying in wrong order
• Using upper matrix incorrectly
• Multiplying by transpose instead of L