Qr Factorisation Calculator

Easily decompose your 2×2 or 3×3 matrix into Q (orthogonal) and R (upper triangular) matrices with our QR Factorisation Calculator.

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 matrices: an orthogonal matrix Q and an upper triangular matrix R. So, A = QR. The QR Factorisation Calculator helps you perform this decomposition for 2×2 or 3×3 matrices.

The matrix Q has orthonormal columns (or rows, depending on the definition, but typically columns), meaning its columns are unit vectors and mutually orthogonal. The matrix R is an upper triangular matrix, meaning all elements below its main diagonal are zero.

This decomposition is widely used in various numerical algorithms, including solving linear systems of equations, finding eigenvalues and eigenvectors (especially in the QR algorithm), and in least squares problems. Anyone working with linear algebra, numerical methods, data analysis, or engineering might use a QR Factorisation Calculator.

A common misconception is that QR factorisation is unique. While the R matrix is unique up to the signs of its diagonal elements (if A is non-singular), the Q matrix can differ in the signs of its columns if the diagonal elements of R are adjusted. However, for a given set of positive diagonal elements in R, Q is unique.

QR Factorisation Formula and Mathematical Explanation

The most common method for computing the QR factorisation is the Gram-Schmidt process, which we’ll illustrate for a 2×2 matrix A = [a1 a2], where a1 and a2 are column vectors.

Let A = [ a11 a12 ]
[ a21 a22 ]. The columns are a1 = [a11, a21]^T and a2 = [a12, a22]^T.

1. First vector:
   • u1 = a1
   • e1 = u1 / ||u1|| (q1 = e1 is the first column of Q)
2. Second vector:
   • u2 = a2 – proj_u1(a2) = a2 – (<a2, u1> / ||u1||²) * u1
   • e2 = u2 / ||u2|| (q2 = e2 is the second column of Q)

So, Q = [e1 e2]. R is then found as R = Q^TA.

R = [ <a1,e1> <a2,e1> ]
[ 0 <a2,e2> ]

For a 3×3 matrix, the process extends similarly with u3 = a3 – proj_u1(a3) – proj_u2(a3).

Variables Table

Variable	Meaning	Unit	Typical Range
A	Input Matrix	Matrix	Real numbers
Q	Orthogonal Matrix	Matrix	Real numbers, columns are orthonormal
R	Upper Triangular Matrix	Matrix	Real numbers, elements below diagonal are zero
aij	Element of matrix A at row i, column j	Scalar	Real numbers
uk	Orthogonal vector before normalization	Vector	Real numbers
ek (or qk)	Orthonormal vector (column of Q)	Vector	Real numbers, ||ek||=1
<v, w>	Inner product (dot product) of vectors v and w	Scalar	Real number
||v||	Norm (magnitude) of vector v	Scalar	Non-negative real number

Our QR Factorisation Calculator implements this process.

Practical Examples (Real-World Use Cases)

Example 1: Solving Linear Systems

Suppose we want to solve Ax = b. If A = QR, then QRx = b. Since Q is orthogonal, Q^TQ = I, so Rx = Q^Tb. This is an upper triangular system, easily solved by back-substitution.

Let A = [ 1 2 ]
[ 3 4 ] and b = [5, 11]^T. Using the QR Factorisation Calculator, we find Q and R.
Then we solve Rx = Q^Tb.

Example 2: Least Squares Problem

In finding the least squares solution to Ax = b where A is m x n (m > n), we solve A^TAx = A^Tb. If A = QR (where Q is m x n with orthonormal columns and R is n x n upper triangular), then R^TQ^TQRx = R^TQ^Tb, which simplifies to R^TRx = R^TQ^Tb, and since R is often invertible, Rx = Q^Tb. The QR Factorisation Calculator provides the Q and R matrices needed here.

For A = [ 1 1 ]
[ 2 1 ]
[ 3 0 ], we would need a rectangular QR, but our calculator focuses on square matrices for full decomposition. However, the principle is similar and often starts with QR of a relevant square matrix or uses specialized methods for rectangular A.

How to Use This QR Factorisation Calculator

1. Select Matrix Size: Choose whether you are entering a 2×2 or 3×3 matrix using the dropdown.
2. Enter Matrix Elements: Input the numerical values for each element (a11, a12, etc.) of your matrix A into the respective fields.
3. Calculate: Click the “Calculate” button (or the results update automatically as you type).
4. View Results: The calculator will display:
   • The orthogonal matrix Q.
   • The upper triangular matrix R.
   • The product Q*R as a verification (it should be close to A, allowing for rounding).
5. Interpret Results: The matrices Q and R are the factors of A such that A = QR. The Q matrix has orthonormal columns, and R is upper triangular. For 2×2 matrices, a visualization of the column vectors of A and Q is also shown.
6. Reset or Copy: Use the “Reset” button to clear inputs to default values or “Copy Results” to copy the Q and R matrices to your clipboard.

This QR Factorisation Calculator is useful for verifying manual calculations or for obtaining quick decompositions.

Key Factors That Affect QR Factorisation Results

• Input Matrix A: The values in matrix A directly determine Q and R. Small changes in A can lead to changes in Q and R.
• Linear Independence of Columns: If the columns of A are linearly dependent (matrix is singular), the Gram-Schmidt process might require careful handling (e.g., if a ui vector becomes zero). Our calculator assumes linearly independent columns for a non-singular R. If columns are near-dependent, numerical stability can be an issue.
• Method Used: While Gram-Schmidt is conceptually simple, other methods like Householder reflections or Givens rotations are often used in practice for better numerical stability, especially for larger or ill-conditioned matrices. Our QR Factorisation Calculator uses Gram-Schmidt.
• Normalization: The normalization step (dividing by the norm) introduces square roots and divisions, which can be sources of numerical error.
• Order of Columns: Changing the order of columns in A will change Q and R, although the underlying space spanned by the columns remains the same.
• Floating-Point Precision: Numerical calculations are done with finite precision, so the resulting Q might not be perfectly orthogonal, and R might not be perfectly upper triangular, and Q*R might only be approximately equal to A. Our QR Factorisation Calculator uses standard JavaScript floating-point arithmetic.

Frequently Asked Questions (FAQ)

Q: Is the QR factorisation of a matrix unique?
A: Not strictly. If A is non-singular, R can be made unique by requiring its diagonal elements to be positive, and then Q is unique. Without this constraint, signs can be flipped between columns of Q and rows of R.

Q: What if the input matrix is singular?
A: If A is singular, at least one of the u_k vectors in Gram-Schmidt might become zero before normalization, or R will have zero(s) on its diagonal. The factorization still exists, but R will be singular. Our calculator handles this but assumes non-zero norms during normalization where possible.

Q: Can I use this QR Factorisation Calculator for non-square matrices?
A: This calculator is designed for square 2×2 and 3×3 matrices. QR factorisation exists for rectangular matrices (m x n), resulting in a Q that is m x m (or m x n) and R that is m x n (or n x n), but this tool is limited to square cases.

Q: What is an orthogonal matrix?
A: A square matrix Q is orthogonal if its columns (and rows) form a set of orthonormal vectors (they are unit vectors and mutually orthogonal). This means Q^TQ = QQ^T = I (the identity matrix), so Q^T = Q^-1.

Q: Why is R upper triangular?
A: The Gram-Schmidt process constructs each q_k to be orthogonal to q₁, …, q_k-1 and as a linear combination of a₁, …, a_k. This structure leads to R = Q^TA being upper triangular.

Q: What are the applications of the QR Factorisation Calculator?
A: It’s used in solving linear systems, least squares fitting, eigenvalue problems (QR algorithm), and data analysis techniques like Principal Component Analysis (PCA) which often involve matrix decompositions.

Q: How accurate is this QR Factorisation Calculator?
A: It uses standard floating-point arithmetic. For well-conditioned matrices, it’s quite accurate. For ill-conditioned matrices, numerical errors can accumulate.

Q: What if I enter non-numeric values?
A: The input fields are of type “number”, but if non-numeric data is forced, the calculation will likely result in NaN (Not a Number) for the outputs. The calculator does basic checks for valid numbers.

Related Tools and Internal Resources

• Matrix Determinant Calculator: Calculate the determinant of 2×2 or 3×3 matrices.
• Matrix Inverse Calculator: Find the inverse of a given square matrix.
• Eigenvalue and Eigenvector Calculator: Compute eigenvalues and eigenvectors for 2×2 matrices.
• Linear Equation Solver: Solve systems of linear equations using various methods.
• Vector Dot Product Calculator: Calculate the dot product of two vectors.
• Gram-Schmidt Orthonormalization Calculator: Step-by-step orthogonalization of vectors.