QR decomposition in SciPy - Step-by-Step Execution

Concept Flow - QR decomposition

Start with matrix A

↓

Apply QR decomposition

↓

Get Q (orthogonal matrix)

↓

Get R (upper triangular matrix)

↓

Verify A = Q * R

↓

End

We start with a matrix A, then split it into Q and R matrices using QR decomposition, and finally verify that multiplying Q and R returns the original matrix.

Execution Sample

SciPy

import numpy as np
from scipy.linalg import qr

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

This code computes the QR decomposition of matrix A and prints matrices Q and R.

Execution Table

Step	Action	Matrix/Variable	Value/Result
1	Define matrix A	A	[[1 2] [3 4]]
2	Apply qr(A)	Q	[[-0.31622777 -0.9486833 ] [-0.9486833 0.31622777]]
3	Apply qr(A)	R	[[-3.16227766 -4.42718872] [ 0. 0.63245553]]
4	Verify A = Q * R	Q @ R	[[1. 2.] [3. 4.]]
5	Print Q	Output	[[-0.31622777 -0.9486833 ] [-0.9486833 0.31622777]]
6	Print R	Output	[[-3.16227766 -4.42718872] [ 0. 0.63245553]]
7	End		QR decomposition complete

💡 QR decomposition finished and verified that Q*R equals original matrix A

Variable Tracker

Variable	Start	After qr(A)	Final
A	[[1 2] [3 4]]	[[1 2] [3 4]]	[[1 2] [3 4]]
Q	None	[[-0.31622777 -0.9486833 ] [-0.9486833 0.31622777]]	[[-0.31622777 -0.9486833 ] [-0.9486833 0.31622777]]
R	None	[[-3.16227766 -4.42718872] [ 0. 0.63245553]]	[[-3.16227766 -4.42718872] [ 0. 0.63245553]]

Key Moments - 3 Insights

Why are the values in Q negative?

The QR decomposition returns an orthogonal matrix Q which can have negative signs due to normalization; the important part is that Q is orthogonal, not that values are positive. See execution_table rows 2 and 5.

Why does R have zeros below the diagonal?

R is an upper triangular matrix by definition in QR decomposition, so all elements below the diagonal are zero. This is shown in execution_table row 3.

How do we know Q and R multiply back to A?

Row 4 in execution_table shows the multiplication Q @ R equals the original matrix A, confirming the decomposition is correct.

Visual Quiz - 3 Questions

Look at the execution_table at step 3, what is the shape of matrix R?

A 2 rows and 3 columns

B 2 rows and 2 columns

C 3 rows and 2 columns

D 1 row and 2 columns

💡 Hint

Check the printed matrix R in execution_table row 3; it has two rows and two columns.

At which step does the code verify that Q times R equals A?

A Step 2

B Step 6

C Step 4

D Step 7

💡 Hint

Look at execution_table row 4 where Q @ R is computed and compared to A.

If matrix A was changed to [[2, 0], [0, 2]], how would Q likely change?

A Q would be the identity matrix

B Q would be all zeros

C Q would be the same as before

D Q would be a zero matrix with negative values

💡 Hint

For a diagonal matrix like [[2,0],[0,2]], Q is usually the identity matrix as per QR decomposition properties.

Concept Snapshot

QR decomposition splits a matrix A into Q and R matrices.
Q is orthogonal (Q.T @ Q = I).
R is upper triangular.
A = Q * R exactly.
Use scipy.linalg.qr to compute.
Useful for solving linear systems and least squares.

Full Transcript

QR decomposition takes a matrix A and breaks it into two matrices Q and R. Q is orthogonal, meaning its transpose times itself is the identity matrix. R is upper triangular, meaning all elements below the diagonal are zero. We use scipy's qr function to get Q and R. After decomposition, multiplying Q and R returns the original matrix A, confirming the process. This is useful in many data science tasks like solving equations and regression.

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