QR Factorisation Calculator

What is QR Factorisation?

QR Factorisation, also known as QR decomposition, is a fundamental process in linear algebra where a matrix A is decomposed into the product of two matrices: an orthogonal matrix Q and an upper triangular matrix R. This can be expressed as A = QR. It's a powerful tool used across various scientific and engineering disciplines.

An **orthogonal matrix Q** is a square matrix whose columns are orthonormal vectors, meaning they are mutually orthogonal (their dot product is zero) and each has a magnitude (Euclidean norm) of one. Consequently, its transpose is equal to its inverse (Q^TQ = I, where I is the identity matrix).

An **upper triangular matrix R** is a square matrix where all the entries below the main diagonal are zero.

Who should use it? QR factorisation is indispensable for numerical analysts, engineers, data scientists, and anyone working with systems of linear equations, least squares solutions, eigenvalue calculations, and singular value decomposition (SVD). It provides a numerically stable method for solving various matrix problems.

QR Factorisation Formula and Explanation

The core formula for QR factorisation is simple: A = QR. The challenge lies in computing Q and R. One of the most intuitive methods for deriving Q and R is the **Gram-Schmidt process**.

Given a matrix A with linearly independent columns a₁, a₂, ..., a_n, the Gram-Schmidt process constructs an orthonormal basis q₁, q₂, ..., q_n for the column space of A. These orthonormal vectors then form the columns of the matrix Q.

The process works as follows:

1. For the first column: q₁ = a₁ / ||a₁||
2. For subsequent columns (j = 2, ..., n):
   • Calculate the projection of a_j onto the previously found orthonormal vectors:
     proj_q a_j = (a_j · q₁)q₁ + (a_j · q₂)q₂ + ... + (a_j · q_j-1)q_j-1
   • Subtract this projection from a_j to get an orthogonal vector v_j:
     v_j = a_j - proj_q a_j
   • Normalize v_j to get q_j:
     q_j = v_j / ||v_j||

Once Q is constructed, the upper triangular matrix R can be found by R = Q^TA. Alternatively, the elements of R can be directly computed during the Gram-Schmidt process:

• r_ii = ||v_i|| (before normalization)
• r_ij = a_j · q_i (for i < j)
• r_ij = 0 (for i > j)

Variables in QR Factorisation

Practical Examples of QR Factorisation

Let's illustrate the qr factorisation calculator with a couple of examples.

Example 1: A Simple 2x2 Matrix

Consider the matrix A:

A = [[1, 2],
     [3, 4]]

Inputs:

• Matrix A Rows: 2
• Matrix A Columns: 2
• Elements: A₁₁=1, A₁₂=2, A₂₁=3, A₂₂=4

Results (approximate):

Q = [[-0.3162,  0.9487],
     [-0.9487, -0.3162]]

R = [[-3.1623, -4.4272],
     [ 0.0000, -0.6325]]

Notice that Q is orthogonal (Q^TQ ≈ I) and R is upper triangular. The verification error (||A - QR||F) and orthogonality check (||Q^TQ - I||F) will be very close to zero, indicating a successful factorisation.

Example 2: A 3x2 (Non-Square) Matrix

QR factorisation is also applicable to non-square matrices. Let's take:

A = [[1, 1],
     [1, 2],
     [1, 3]]

Inputs:

• Matrix A Rows: 3
• Matrix A Columns: 2
• Elements: A₁₁=1, A₁₂=1, A₂₁=1, A₂₂=2, A₃₁=1, A₃₂=3

Results (approximate):

Q = [[ 0.5774, -0.8165],
     [ 0.5774, -0.0000],
     [ 0.5774,  0.8165]]

R = [[1.7321, 3.4641],
     [0.0000, 0.8165]]

In this case, Q is a 3x2 matrix with orthonormal columns, and R is a 2x2 upper triangular matrix. The product QR will yield the original 3x2 matrix A. This is particularly useful for solving least squares problems where A is often tall and skinny.

How to Use This QR Factorisation Calculator

Our qr factorisation calculator is designed for simplicity and accuracy. Follow these steps to get your matrix decomposition:

1. Enter Matrix Dimensions: Start by specifying the number of rows (m) and columns (n) for your matrix A in the respective input fields. The calculator currently supports matrices up to 10x10.
2. Input Matrix Elements: After setting the dimensions, a grid of input fields will appear. Enter the numerical values for each element of your matrix A. Ensure all entries are valid numbers (integers or decimals).
3. Click "Calculate QR Factorisation": Once all elements are entered, click the "Calculate QR Factorisation" button. The calculator will process the matrix using the Gram-Schmidt algorithm.
4. Interpret Results:
   • Orthogonal Matrix Q: This matrix will be displayed. Its columns are orthonormal vectors.
   • Upper Triangular Matrix R: This matrix will also be displayed. All values below its main diagonal will be zero.
   • Primary Highlighted Result: A brief summary confirming the factorisation.
   • Verification Error (||A - QR||F): This value shows the Frobenius norm of the difference between your original matrix A and the product of Q and R. A value very close to zero (e.g., 1e-15) indicates a highly accurate factorisation.
   • Orthogonality Check (||QᵀQ - I||F): This value shows how close the matrix Q is to being perfectly orthogonal. Again, a value very close to zero signifies excellent orthogonality.
   • Factorisation Quality Chart: A visual representation of the two error metrics, helping you quickly gauge the accuracy.
5. Copy Results: Use the "Copy Results" button to easily transfer the calculated Q and R matrices, along with the error checks, to your clipboard for documentation or further use.
6. Reset: If you wish to start over with a new matrix, click the "Reset" button to clear all inputs and results.

Since matrix elements are unitless, there is no unit switcher required. All calculations are performed on raw numerical values.

Key Factors That Affect QR Factorisation

While the concept of QR factorisation is straightforward, several factors can influence its computation and utility, especially in practical numerical applications:

1. Matrix Dimensions (m x n): The size of the matrix directly impacts the computational cost. Larger matrices require significantly more operations. For this qr factorisation calculator, we limit dimensions to prevent performance issues in a browser environment.
2. Linear Independence of Columns: For the Gram-Schmidt process to work effectively, the columns of matrix A should be linearly independent. If columns are linearly dependent, the process might encounter division by zero (if a vector becomes zero) or produce numerically unstable results.
3. Condition Number of A: A matrix's condition number indicates its sensitivity to input changes. Matrices with high condition numbers are "ill-conditioned," meaning small changes in input can lead to large changes in output. This can affect the numerical stability and accuracy of the QR factorisation.
4. Choice of Algorithm: While this calculator uses the classical Gram-Schmidt process for its conceptual simplicity, other algorithms like modified Gram-Schmidt, Householder reflections, or Givens rotations are often preferred in professional software for their superior numerical stability, especially for ill-conditioned matrices.
5. Floating Point Precision: All computer calculations involve finite precision. This can lead to small errors accumulating during iterative processes, causing the resulting Q matrix to be only "approximately" orthogonal and the R matrix to have very small, but non-zero, values below the diagonal. Our error checks quantify this.
6. Sparsity: If matrix A contains many zero elements (a sparse matrix), specialized QR algorithms can be used to exploit this structure for computational efficiency. Our calculator treats all matrices as dense.

Frequently Asked Questions about QR Factorisation

Q1: What is the primary purpose of QR factorisation?

A1: QR factorisation is primarily used for solving systems of linear equations, finding least squares solutions to overdetermined systems, computing eigenvalues, and performing singular value decomposition (SVD).

Q2: Is the QR factorisation unique?

A2: For a given matrix A, the QR factorisation (A=QR) is unique if we require the diagonal elements of R to be positive. Without this constraint, Q and R are unique up to multiplication by a diagonal matrix with entries ±1.

Q3: What does it mean for a matrix Q to be "orthogonal"?

A3: An orthogonal matrix Q is a square matrix whose columns (and rows) are orthonormal vectors. This means each column vector has a length (norm) of 1, and any two distinct column vectors are orthogonal (their dot product is 0). A key property is that Q^TQ = I (the identity matrix), which also implies Q^T = Q^-1.

Q4: What is an "upper triangular matrix" R?

A4: An upper triangular matrix R is a square matrix where all the entries below the main diagonal are zero. For example, a 3x3 upper triangular matrix would look like: [[a, b, c], [0, d, e], [0, 0, f]].

Q5: Can this qr factorisation calculator handle non-square matrices?

A5: Yes, the qr factorisation calculator can handle non-square matrices (where the number of rows 'm' is not equal to the number of columns 'n'). If m ≥ n, the factorisation A = QR still holds, where Q is an m × n matrix with orthonormal columns, and R is an n × n upper triangular matrix. If m < n, QR factorisation is typically applied to A^T.

Q6: What is the difference between Gram-Schmidt and Householder reflections for QR factorisation?

A6: Both are methods for QR factorisation. Gram-Schmidt (used in this calculator) constructs Q by orthonormalizing the columns of A sequentially. Householder reflections use a sequence of orthogonal transformations (reflections) to zero out elements below the diagonal, directly forming R and implicitly Q. Householder reflections are generally preferred in numerical computing for their superior numerical stability, especially with ill-conditioned matrices, as they avoid the "loss of orthogonality" that can sometimes occur with classical Gram-Schmidt due to floating-point errors.

Q7: Why is numerical stability important in QR factorisation?

A7: Numerical stability refers to how an algorithm handles errors (like floating-point errors) during computation. For QR factorisation, an unstable algorithm might produce a Q matrix that is not truly orthogonal or an R matrix that is not truly upper triangular, leading to inaccurate results, especially for large or ill-conditioned matrices. This is why methods like Householder transformations are often preferred over classical Gram-Schmidt.

Q8: Does this calculator support complex numbers?

A8: No, this qr factorisation calculator is designed for real-valued matrices only. All input elements must be real numbers (integers or decimals).

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