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.

Practice

1. What does QR decomposition do to a matrix A?

easy

A. Converts A into a lower triangular matrix only

B. Splits A into two diagonal matrices

C. Splits A into an orthogonal matrix Q and an upper triangular matrix R

D. Finds the inverse of A directly

Solution

1. Step 1: Understand QR decomposition definition

   QR decomposition splits a matrix into two parts: an orthogonal matrix Q and an upper triangular matrix R.

2. Step 2: Compare options with definition

   Only Splits A into an orthogonal matrix Q and an upper triangular matrix R correctly describes this split. Others describe different matrix operations.

3. Final Answer:

   Splits A into an orthogonal matrix Q and an upper triangular matrix R -> Option C

4. Quick Check:

   QR decomposition = Q and R matrices [OK]

Hint: QR always produces Q (orthogonal) and R (upper triangular) [OK]

Common Mistakes:

• Confusing QR with LU decomposition
• Thinking QR produces diagonal matrices
• Assuming QR finds matrix inverse

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

easy

A. from scipy.linalg import qr
Q, R = qr(A)

B. import scipy.qr
Q, R = scipy.qr(A)

C. from scipy import qr
Q, R = qr(A)

D. import qr from scipy.linalg
Q, R = qr(A)

Solution

1. Step 1: Recall scipy QR import syntax

   The correct import is from scipy.linalg import qr.

2. Step 2: Check usage correctness

   Calling qr(A) returns Q and R. from scipy.linalg import qr Q, R = qr(A) matches this exactly.

3. Final Answer:

   from scipy.linalg import qr Q, R = qr(A) -> Option A

4. Quick Check:

   Correct import and call = from scipy.linalg import qr Q, R = qr(A) [OK]

Hint: Use 'from scipy.linalg import qr' to access QR function [OK]

Common Mistakes:

• Trying to import qr directly from scipy
• Using incorrect import syntax
• Calling qr without importing

3. Given the matrix A = [[1, 2], [3, 4]], what is the shape of Q and R after Q, R = qr(A)?

medium

A. Q is (2, 2) and R is (2, 1)

B. Q is (1, 2) and R is (2, 2)

C. Q is (2, 1) and R is (1, 2)

D. Q is (2, 2) and R is (2, 2)

Solution

1. Step 1: Check input matrix shape

   Matrix A has shape (2, 2).

2. Step 2: Understand QR output shapes

   For a square matrix, Q and R both have the same shape as A, which is (2, 2).

3. Final Answer:

   Q is (2, 2) and R is (2, 2) -> Option D

4. Quick Check:

   Square matrix QR outputs same shape matrices [OK]

Hint: Q and R shapes match input matrix shape for square matrices [OK]

Common Mistakes:

• Assuming R has fewer columns
• Confusing Q shape with R shape
• Thinking Q or R is 1D

4. What is wrong with this code snippet for QR decomposition?

from scipy.linalg import qr
A = [[1, 2], [3, 4]]
Q, R = qr(A)
print(Q)

medium

A. Nothing is wrong with this code snippet

B. The qr function does not return two values

C. Matrix A should be a numpy array, not a list of lists

D. Import statement is incorrect

Solution

1. Step 1: Verify import statement

   The import from scipy.linalg import qr is correct.

2. Step 2: Check qr function returns

   qr(A) returns two values Q and R.

3. Step 3: Validate input type

   List of lists for A works as qr accepts array-like inputs and converts internally.

4. Step 4: Check print statement

   print(Q) correctly outputs the Q matrix.

5. Final Answer:

   Nothing is wrong with this code snippet -> Option A

6. Quick Check:

   Code runs successfully, produces correct Q and R [OK]

Hint: scipy.linalg.qr accepts lists of lists as array-like input [OK]

Common Mistakes:

• Thinking Python lists cannot be passed to qr
• Misunderstanding qr return values
• Incorrect import statements

5. You want to solve the system Ax = b using QR decomposition where A is a 3x3 matrix and b is a vector. Which step correctly describes how to find x after computing Q, R = qr(A)?

hard

A. Compute x = Q @ b then solve Ry = x

B. Compute y = Q.T @ b then solve Rx = y using back substitution

C. Compute y = R @ b then solve Qx = y

D. Compute x = R.T @ Q @ b

Solution

1. Step 1: Use QR decomposition to rewrite Ax = b

   Since A = QR, rewrite as QRx = b.

2. Step 2: Multiply both sides by Q.T

   Multiply by Q.T to get Rx = Q.T b. Define y = Q.T b.

3. Step 3: Solve upper triangular system

   Now solve Rx = y using back substitution to find x.

4. Final Answer:

   Compute y = Q.T @ b then solve Rx = y using back substitution -> Option B

5. Quick Check:

   Use Q.T b and solve Rx = y [OK]

Hint: Multiply b by Q.T, then solve Rx = y [OK]

Common Mistakes:

• Multiplying b by Q instead of Q.T
• Trying to solve Qx = y instead of Rx = y
• Ignoring that R is upper triangular