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.