QR decomposition in SciPy - Mini Project: Build & Apply

QR Decomposition with SciPy

📖 Scenario: You work as a data analyst and need to break down a matrix into simpler parts to understand its structure better. QR decomposition is a method that helps split a matrix into two parts: one with orthogonal columns and one upper triangular matrix. This is useful in solving systems of equations and data fitting.

🎯 Goal: Build a Python program that uses SciPy to perform QR decomposition on a given matrix and display the resulting matrices.

📋 What You'll Learn

Create a 3x3 matrix using NumPy with exact values

Set up a configuration variable to choose the mode of QR decomposition

Use SciPy's qr function to decompose the matrix

Print the Q and R matrices as output

💡 Why This Matters

🌍 Real World

QR decomposition is used in data fitting, solving linear systems, and simplifying matrix computations in engineering and science.

💼 Career

Understanding QR decomposition helps in roles like data analyst, data scientist, and engineer where matrix operations and numerical methods are common.

1

Create the matrix

Import NumPy as np and create a 3x3 matrix called A with these exact values: [[1, 2, 3], [4, 5, 6], [7, 8, 9]].

SciPy

import numpy as np
# Create the matrix A with the given values
# Your code here

Hint

Use np.array to create the matrix with the exact values.

2

Set QR decomposition mode

Create a variable called mode and set it to the string 'reduced' to specify the QR decomposition mode.

SciPy

import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
# Set the mode variable to 'reduced'
# Your code here

Hint

Assign the string 'reduced' to the variable mode.

3

Perform QR decomposition

Import the qr function from scipy.linalg and use it to decompose matrix A with the mode set by mode. Store the results in variables Q and R.

SciPy

import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
mode = 'reduced'
# Import qr from scipy.linalg and perform QR decomposition
# Your code here

Hint

Use from scipy.linalg import qr and call qr(A, mode=mode).

4

Display the results

Print the matrices Q and R on separate lines using print statements.

SciPy

import numpy as np
A = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
mode = 'reduced'
from scipy.linalg import qr
Q, R = qr(A, mode=mode)
# Print Q and R matrices
# Your code here

Hint

Use two print statements: one for Q and one for R.

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