Qr Factorisation Calculator

Calculate QR Factorisation

Enter the elements of a 2×2 matrix A to find its QR factorisation (A = QR), where Q is an orthogonal matrix and R is an upper triangular matrix.

What is QR Factorisation?

QR Factorisation (also known as QR decomposition) is a fundamental concept in linear algebra where a matrix A is decomposed into a product of two other matrices: an orthogonal matrix Q and an upper triangular matrix R. So, A = QR. The QR factorisation calculator helps perform this decomposition.

An orthogonal matrix Q has the property that its transpose is equal to its inverse (Q^TQ = I), meaning its columns (and rows) form an orthonormal set of vectors. An upper triangular matrix R has all its elements below the main diagonal equal to zero. The QR factorisation is widely used in numerical linear algebra, for example, to solve linear systems of equations, find eigenvalues and eigenvectors (through the QR algorithm), and in least-squares problems. Our QR factorisation calculator uses the Gram-Schmidt process for this.

Who should use it?

Students, engineers, data scientists, and anyone working with linear algebra, matrix operations, or numerical methods will find a QR factorisation calculator useful. It’s particularly helpful for understanding the Gram-Schmidt process and its application in decomposing matrices.

Common misconceptions

A common misconception is that the QR factorisation is unique. While the decomposition A=QR is unique up to the signs of the columns of Q (and corresponding rows of R) if A is non-singular, different methods or sign conventions can lead to slightly different Q and R matrices that still satisfy A=QR and the properties of Q and R. Our QR factorisation calculator provides one valid decomposition.

QR Factorisation Formula and Mathematical Explanation

The QR factorisation of a matrix A can be obtained using several methods, with the Gram-Schmidt process being one of the most intuitive for understanding. For a matrix A with linearly independent columns a₁, a₂, …, a_n, we construct orthonormal vectors q₁, q₂, …, q_n that form the columns of Q.

For a 2×2 matrix A = [a₁|a₂] = [[a₁₁, a₂₁]^T | [a₁₂, a₂₂]^T]:

1. First column:
   • u₁ = a₁
   • r₁₁ = ||u₁|| = √(a₁₁² + a₂₁²)
   • q₁ = u₁ / r₁₁
2. Second column:
   • u₂ = a₂ – proj_u1(a₂) = a₂ – (q₁·a₂)q₁ (since q₁ is normalized)
   • r₁₂ = q₁·a₂
   • r₂₂ = ||u₂||
   • q₂ = u₂ / r₂₂

So, Q = [q₁|q₂] and R = [[r₁₁, r₁₂], [0, r₂₂]]. The QR factorisation calculator implements this process.

Variables Table

Variable	Meaning	Unit	Typical Range
A	Input Matrix	Dimensionless	Real numbers
Q	Orthogonal Matrix	Dimensionless	Elements between -1 and 1
R	Upper Triangular Matrix	Dimensionless	Real numbers
aij	Elements of A	Dimensionless	Real numbers
qij	Elements of Q	Dimensionless	Real numbers
rij	Elements of R	Dimensionless	Real numbers
ui	Intermediate orthogonal vectors	Dimensionless	Real numbers
||ui||	Magnitude (norm) of ui	Dimensionless	Non-negative real numbers

Practical Examples (Real-World Use Cases)

Example 1: Solving a Linear System

Suppose we want to solve Ax = b. If we have A = QR, then QRx = b. Since Q is orthogonal, Q^TQ = I, so Rx = Q^Tb. This transforms the original system into an equivalent system with an upper triangular matrix R, which is easy to solve using back-substitution. The QR factorisation calculator gives you Q and R.

Let A = [[1, 2], [3, 4]] (as in the default calculator values).
The QR factorisation calculator gives:
Q ≈ [[0.316, 0.949], [0.949, -0.316]]
R ≈ [[3.162, 4.427], [0, 0.632]]

Example 2: Least Squares Problem

In least squares problems, we often want to minimize ||Ax – b||². Using QR factorisation, A=QR, we minimize ||QRx – b||² = ||Rx – Q^Tb||² (since Q preserves norms). This is minimized when Rx = Q^Tb, again solvable by back-substitution.

Let A = [[1, 1], [1, 2], [1, 3]] (a 3×2 matrix – our calculator is 2×2, but the principle is the same). We would first find the QR factorisation of A.

How to Use This QR Factorisation Calculator

1. Enter Matrix A Elements: Input the values for the 2×2 matrix A into the fields A(1,1), A(1,2), A(2,1), and A(2,2).
2. Calculate: Click the “Calculate” button or simply change any input value. The QR factorisation calculator will automatically compute the Q and R matrices.
3. View Results: The calculator displays:
   • The elements of the orthogonal matrix Q.
   • The elements of the upper triangular matrix R.
   • Intermediate values like ||u₁|| and ||u₂||.
   • A bar chart showing the magnitudes ||u₁|| and ||u₂||.
4. Reset: Click “Reset” to return to the default matrix values.
5. Copy Results: Click “Copy Results” to copy the Q and R matrix elements and intermediate values to your clipboard.

The QR factorisation calculator provides a quick way to perform the decomposition and visualize some intermediate steps.

Key Factors That Affect QR Factorisation Results

1. Input Matrix A: The values in matrix A directly determine Q and R.
2. Linear Independence of Columns: The Gram-Schmidt process (and thus QR factorisation) works best when the columns of A are linearly independent. If they are linearly dependent, one of the r_ii values will be zero, and the process might need modification or indicates A is singular.
3. Numerical Stability: For matrices that are nearly singular or have columns close to being linearly dependent, the standard Gram-Schmidt process can be numerically unstable. Modified Gram-Schmidt or Householder reflections are more stable methods for QR factorisation (our calculator uses standard Gram-Schmidt for simplicity).
4. Choice of Orthonormalisation Method: While our QR factorisation calculator uses Gram-Schmidt, other methods like Householder reflections or Givens rotations also yield a QR factorisation, though the Q matrix might differ in signs.
5. Precision: The precision of the input numbers and the calculations affects the accuracy of the resulting Q and R matrices. Small rounding errors can accumulate.
6. Matrix Dimensions: While this calculator is for 2×2, the concept of QR factorisation extends to m x n matrices. The complexity increases with size.

Frequently Asked Questions (FAQ)

What is QR factorisation used for?

QR factorisation is used in solving linear systems, least-squares problems, eigenvalue calculations (QR algorithm), and in various numerical algorithms. Check our solving linear systems guide.

Is the QR factorisation unique?

For a non-singular matrix A, the QR factorisation A=QR is unique up to the signs of the columns of Q if we require the diagonal elements of R to be positive. Different sign conventions can be adopted.

What if the columns of A are linearly dependent?

If the columns of A are linearly dependent, at some stage in the Gram-Schmidt process, a vector u_i will be zero, and ||u_i|| (which becomes r_ii) will be zero. This indicates A is singular. Our basic QR factorisation calculator might show ||u2||=0 in such cases.

What is an orthogonal matrix?

An orthogonal matrix Q is a square matrix whose columns and rows are orthonormal vectors (they are unit vectors and orthogonal to each other). This means Q^TQ = QQ^T = I, and Q^T = Q^-1. See more on orthogonal vectors explained.

What is an upper triangular matrix?

An upper triangular matrix R is a square matrix where all the entries below the main diagonal are zero (r_ij = 0 for i > j).

Can I use this QR factorisation calculator for 3×3 matrices?

This specific calculator is designed for 2×2 matrices for simplicity of input. The principle extends to larger matrices, but the manual input and calculation display become more complex.

How does this relate to the Gram-Schmidt process?

The QR factorisation is essentially the matrix form of the Gram-Schmidt process of orthonormalising a set of vectors (the columns of A).

What are other matrix decompositions?

Other common matrix decompositions include LU decomposition, Singular Value Decomposition (SVD), and Cholesky decomposition, each with different properties and applications. You might be interested in our eigenvalue calculator.

Related Tools and Internal Resources

• Gram-Schmidt Calculator: Perform the Gram-Schmidt orthonormalisation process step-by-step.
• Eigenvalue Calculator: Find eigenvalues and eigenvectors for matrices, often using methods related to QR factorisation (like the QR algorithm).
• Matrix Multiplication Calculator: Multiply matrices, which can be useful to verify A = QR after using the QR factorisation calculator.
• Linear Algebra Basics: Learn the fundamental concepts of linear algebra that underpin QR factorisation.
• Solving Linear Systems: Understand how matrix decompositions like QR factorisation help in solving systems of linear equations.
• Orthogonal Vectors Explained: A guide to understanding orthogonal and orthonormal vectors, key to the Q matrix in QR factorisation.