Cholesky decomposition in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Cholesky decomposition with scipy

What is the output of the following code that performs Cholesky decomposition on a positive definite matrix?

SciPy

import numpy as np
from scipy.linalg import cholesky

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

A

[[4. 2.]
 [2. 3.]]

B

[[2. 1.]
 [0. 1.41421356]]

C

[[2. 0.]
 [2. 1.41421356]]

D

[[2. 0.]
 [1. 1.41421356]]

💡 Hint

Remember that the Cholesky decomposition returns a lower triangular matrix L such that A = L * L.T

The matrix A is decomposed into L where L is lower triangular. The first element is sqrt(4) = 2, and the (2,1) element is 2/2 = 1. The (2,2) element is sqrt(3 - 1^2) = sqrt(2) ≈ 1.4142.

❓ data_output

intermediate

Shape of Cholesky factor matrix

Given a 3x3 positive definite matrix, what is the shape of the matrix returned by scipy.linalg.cholesky with lower=True?

SciPy

import numpy as np
from scipy.linalg import cholesky

A = np.array([[6, 3, 4], [3, 6, 5], [4, 5, 10]])
L = cholesky(A, lower=True)
print(L.shape)

A (3, 1)

B (3, 3)

C (1, 3)

D (6, 6)

💡 Hint

The Cholesky decomposition returns a matrix of the same size as the input matrix.

The Cholesky factor L has the same dimensions as the original matrix A, so it is 3x3.

🔧 Debug

advanced

Error raised by Cholesky on non-positive definite matrix

What error does the following code raise when trying to compute the Cholesky decomposition of a non-positive definite matrix?

SciPy

import numpy as np
from scipy.linalg import cholesky

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

A numpy.linalg.LinAlgError: Matrix is not positive definite

B ValueError: shapes (2,2) and (2,2) not aligned

C TypeError: unsupported operand type(s) for /: 'int' and 'str'

D No error, returns a valid matrix

💡 Hint

Cholesky decomposition requires the matrix to be positive definite.

The matrix is symmetric but not positive definite, so scipy.linalg.cholesky raises a LinAlgError indicating the matrix is not positive definite.

🚀 Application

advanced

Using Cholesky decomposition to solve linear systems

Which code snippet correctly uses Cholesky decomposition to solve the system Ax = b for x?

A

import numpy as np
from scipy.linalg import cholesky

A = np.array([[4, 2], [2, 3]])
b = np.array([6, 8])
L = cholesky(A, lower=True)
x = np.dot(L, b)
print(x)

B

import numpy as np
from scipy.linalg import cholesky

A = np.array([[4, 2], [2, 3]])
b = np.array([6, 8])
L = cholesky(A)
x = np.linalg.solve(L, b)
print(x)

C

import numpy as np
from scipy.linalg import cholesky, solve_triangular

A = np.array([[4, 2], [2, 3]])
b = np.array([6, 8])
L = cholesky(A, lower=True)
y = solve_triangular(L, b, lower=True)
x = solve_triangular(L.T, y, lower=False)
print(x)

D

import numpy as np
from scipy.linalg import cholesky

A = np.array([[4, 2], [2, 3]])
b = np.array([6, 8])
L = cholesky(A, lower=True)
x = np.linalg.inv(L).dot(b)
print(x)

💡 Hint

Use solve_triangular twice: first for Ly = b, then for L.T x = y.

Cholesky decomposition gives A = L L.T. To solve Ax = b, solve Ly = b for y, then L.T x = y for x using solve_triangular.

🧠 Conceptual

expert

Cholesky decomposition properties and usage

Which statement about Cholesky decomposition is TRUE?

A Cholesky decomposition requires the matrix to be symmetric and positive definite.

B The Cholesky factor L is always an upper triangular matrix.

C Cholesky decomposition returns two matrices L and U such that A = L * U.

D Cholesky decomposition can be applied to any square matrix, regardless of symmetry or definiteness.

💡 Hint

Think about the conditions on the matrix for Cholesky to work.

Cholesky decomposition only works on symmetric, positive definite matrices and returns a lower (or upper) triangular matrix L such that A = L L.T.