QR decomposition in SciPy - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is QR decomposition in simple terms?

QR decomposition breaks a matrix into two parts: Q, which is an orthogonal matrix (like a set of perpendicular directions), and R, which is an upper triangular matrix (like a staircase shape). This helps solve equations and understand matrix properties.

beginner

What does the Q matrix represent in QR decomposition?

Q is an orthogonal matrix, meaning its columns are at right angles to each other and have length 1. This means Q's transpose is also its inverse, which is useful for calculations.

beginner

What is the shape and property of the R matrix in QR decomposition?

R is an upper triangular matrix, which means all the numbers below the diagonal are zero. It is the part that contains the 'weights' or 'scaling' in the decomposition.

intermediate

How can QR decomposition help solve linear systems?

Instead of solving Ax = b directly, we write A as QR. Then solve Qy = b (easy because Q is orthogonal), and then solve Rx = y (easy because R is triangular). This is more stable and efficient.

beginner

Which scipy function is used for QR decomposition?

The function scipy.linalg.qr() computes the QR decomposition of a matrix. It returns Q and R matrices that you can use for further calculations.

What type of matrix is Q in QR decomposition?

A Diagonal matrix

B Symmetric matrix

C Lower triangular matrix

D Orthogonal matrix

Q is an orthogonal matrix, meaning its columns are perpendicular and have length 1.

What shape does the R matrix have in QR decomposition?

A Diagonal

B Lower triangular

C Upper triangular

D Symmetric

R is an upper triangular matrix, with zeros below the diagonal.

Which scipy function is used to perform QR decomposition?

A scipy.linalg.eig()

B scipy.linalg.qr()

C scipy.linalg.svd()

D scipy.linalg.inv()

scipy.linalg.qr() computes the QR decomposition.

Why is QR decomposition useful for solving Ax = b?

A Because Q is orthogonal and R is triangular, making the system easier to solve

B Because R is orthogonal

C Because Q is diagonal

D Because it reduces matrix size

Q being orthogonal and R triangular simplifies solving the system step-by-step.

If A = QR, what is the product Q * R equal to?

A Matrix A

B Transpose of A

C Zero matrix

D Identity matrix

The product Q * R reconstructs the original matrix A.

Explain the steps to solve a linear system Ax = b using QR decomposition.

Think about breaking the problem into simpler parts using Q and R.

Describe the properties of Q and R matrices in QR decomposition and why they are useful.

Focus on shape and orthogonality.

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