QR decomposition in SciPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of QR decomposition matrices shapes

Given the following code using scipy.linalg.qr, what are the shapes of matrices Q and R?

SciPy

import numpy as np
from scipy.linalg import qr

A = np.array([[1, 2], [3, 4], [5, 6]])
Q, R = qr(A, mode='economic')
print(Q.shape, R.shape)

A (2, 2) (2, 2)

B (3, 3) (3, 2)

C (3, 2) (3, 3)

D (3, 2) (2, 2)

💡 Hint

The 'economic' mode returns reduced shapes for Q and R.

For a 3x2 matrix A, economic QR returns Q with shape (3, 2) and R with shape (2, 2).

❓ data_output

intermediate

Result of reconstructing matrix from QR factors

What is the output of the following code that reconstructs matrix A from its QR decomposition?

SciPy

import numpy as np
from scipy.linalg import qr

A = np.array([[2, 0], [1, 3]])
Q, R = qr(A)
reconstructed = np.dot(Q, R)
print(np.round(reconstructed, decimals=5))

A

[[2. 0.]
 [1. 3.]]

B

[[2. 0.]
 [0. 3.]]

C

[[2. 1.]
 [0. 3.]]

D

[[1. 0.]
 [1. 3.]]

💡 Hint

Multiplying Q and R should give back the original matrix A.

The product Q*R reconstructs the original matrix A exactly (within numerical precision).

🔧 Debug

advanced

Identify the error in QR decomposition code

What error will this code raise when executed?

SciPy

import numpy as np
from scipy.linalg import qr

A = np.array([[1, 2], [3, 4]])
Q, R = qr(A, mode='invalid_mode')

A AttributeError: module 'scipy.linalg' has no attribute 'qr'

B TypeError: qr() missing required positional argument

C ValueError: Unknown mode 'invalid_mode'

D No error, code runs successfully

💡 Hint

Check the valid modes for the qr function.

The mode parameter only accepts specific strings like 'full', 'economic', 'r', or 'raw'. 'invalid_mode' causes a ValueError.

🧠 Conceptual

advanced

Properties of Q matrix from QR decomposition

Which statement about the Q matrix from QR decomposition is true?

A Q is always a diagonal matrix

B Q is an orthogonal matrix with Q.T @ Q = I

C Q is a lower triangular matrix

D Q is the inverse of R

💡 Hint

Think about what orthogonal means for matrices.

Q is orthogonal, meaning its transpose times itself equals the identity matrix.

🚀 Application

expert

Using QR decomposition to solve linear system

Given matrix A and vector b, which code correctly solves Ax = b using QR decomposition?

SciPy

import numpy as np
from scipy.linalg import qr

A = np.array([[1, 2], [3, 4]])
b = np.array([5, 11])

A

Q, R = qr(A)
x = np.linalg.solve(R, Q.T @ b)
print(x)

B

Q, R = qr(A)
x = np.linalg.solve(Q, R @ b)
print(x)

C

Q, R = qr(A)
x = np.dot(R, np.linalg.inv(Q)) @ b
print(x)

D

Q, R = qr(A)
x = np.dot(np.linalg.inv(R), np.dot(Q, b))
print(x)

💡 Hint

Recall that A = QR and solve Rx = Q^T b for x.

Since A=QR, Ax=b becomes QRx=b, so Rx=Q^Tb. Solve Rx=Q^Tb for x.