QR decomposition in SciPy - Step-by-Step Execution

Concept Flow - QR decomposition

Start with matrix A

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Apply QR decomposition

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Get Q (orthogonal matrix)

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Get R (upper triangular matrix)

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Verify A = Q * R

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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.