QR Decomposition Calculator

Decompose a matrix into its orthogonal matrix Q and upper triangular matrix R with QR Decomposition Calculator using the Gram-Schmidt process

Calculate QR Decomposition

Number of Rows

Number of Columns

Decimal Precision

Matrix Elements

Example Matrices

Related Calculators →

Mathematics

Eigenvalue Calculator

Calculate the eigenvalues of a matrix upto size 6x6 with our free eigenvalue calculator.

Mathematics

Null Space Calculator

Calculate the null space of a matrix upto size 10x10 with the null space calculator.

Mathematics

RREF Calculator

Calculate the reduced row echelon form of a matrix with our RREF calculator.

QR Decomposition Calculator: Factorize A into QR

Matrix factorizations are cornerstones of numerical linear algebra, breaking down complex matrices into simpler, structured components. The QR decomposition is one such fundamental factorization, expressing a matrix A as the product of an orthogonal matrix Q and an upper triangular matrix R. Manually computing this decomposition, especially for larger matrices, involves intricate algorithms like Gram-Schmidt orthogonalization, Householder reflections, or Givens rotations, which can be computationally intensive and prone to numerical errors.

A QR Decomposition Calculator is a computational tool designed to automate this process. Users input the elements of their matrix (square or rectangular), and the calculator efficiently computes and displays the corresponding Q and R matrices, simplifying tasks in various scientific and engineering disciplines.

QR Decomposition Calculator: Quick Overview

Calculate the QR decomposition of matrices up to 5x5 using the Gram-Schmidt process. Get step-by-step solutions and detailed explanations.

Instant Calculations

Get Q and R matrices instantly using the Gram-Schmidt orthogonalization process

Matrix Operations

Support for matrices up to 5x5 size with linear independence validation

Educational Content

Learn about QR decomposition and its applications

Smart Features

AI-powered explanations and step-by-step solutions

Perfect for students, engineers, and professionals working with linear algebra. Includes detailed examples and real-world applications.

The QR Decomposition Calculator is designed to provide a fast and accurate computation of the Q and R factors for any given matrix A. Whether you are dealing with square or rectangular matrices, this tool simplifies the factorization process, offering a valuable resource for students learning linear algebra, engineers solving practical problems, and researchers working with matrix computations.

Our QR decomposition calculator employs robust numerical methods to ensure the accuracy and stability of the computed decomposition. By handling the complex calculations, it allows users to focus on understanding the properties and applications of QR decomposition in fields ranging from solving linear systems and least-squares problems to eigenvalue computations and data analysis.

What is QR Decomposition?

In linear algebra, the QR decomposition (also known as QR factorization) of a matrix A is a decomposition of the matrix into a product A = QR, where Q is an orthogonal matrix (or unitary matrix if A has complex entries) and R is an upper triangular matrix (also called right triangular matrix). This means Q satisfies $Q^TQ = I$ (or Q*Q = I for unitary), and all entries below the main diagonal of R are zero.

The dimensions of Q and R depend on the dimensions of A (say, m x n) and whether a "full" or "reduced" decomposition is performed:

Full QR Decomposition: If A is m x n, then Q is an m x m orthogonal matrix, and R is an m x n upper trapezoidal matrix (meaning it's upper triangular if m ≤ n, and its bottom m-n rows are all zeros if m > n).
Reduced (Thin) QR Decomposition: If A is m x n with m ≥ n and has full column rank, then Q is an m x n matrix with orthonormal columns $Q^TQ = I_n$ , and R is an n x n invertible upper triangular matrix. This is often more computationally efficient and sufficient for many applications like solving least-squares problems.

A_{m \times n} = Q_{m \times m} R_{m \times n} \quad (Full \ QR)

A_{m \times n} = Q_{m \times n} R_{n \times n} \quad (Reduced \ QR, \ for \ m \ge n)

QR decomposition is a cornerstone of numerical algorithms due to the desirable properties of orthogonal and triangular matrices. Orthogonal transformations preserve geometric properties like lengths and angles, contributing to numerical stability, while triangular systems are easy to solve.

Key Properties of QR Decomposition

The QR decomposition possesses several important properties that make it highly useful in numerical computations:

Existence: Every real or complex matrix A (regardless of shape or rank) has a QR decomposition, although it might not be unique.
Orthogonality/Unitarity of Q: The matrix Q has orthonormal columns. For a real matrix A, Q is orthogonal, meaning $Q^TQ = QQ^T = I$ (if Q is square, full QR) or $Q^TQ = I$ (if Q is rectangular, reduced QR). This implies that multiplication by Q preserves the Euclidean norm ${||Q_x||}_2 = {||x||}_2$ and angles between vectors.
Upper Triangularity of R: The matrix R is upper triangular (or upper trapezoidal in the full QR case), meaning all elements below the main diagonal are zero. This structure makes solving systems of equations involving R straightforward using back-substitution.
Uniqueness: For a full-rank square matrix A, the QR decomposition is unique if we impose the condition that the diagonal elements of R are positive. For rectangular or rank-deficient matrices, uniqueness conditions are more complex or may not hold without further constraints.
Relation to Column Space: The first k columns of Q form an orthonormal basis for the subspace spanned by the first k columns of A (assuming linear independence holds up to column k). This property is directly linked to the Gram-Schmidt process.
Determinant Relation (for square A): If A is a square matrix, A = QR, then $det{A} = det{Q} \ det{R}$ . Since Q is orthogonal, $det{Q} = \pm 1$ . The determinant of R is simply the product of its diagonal entries. Therefore, $|det{A}| = |det{R}| = |r_{11} * r_{22} * ... * r_{nn}|$ .

Methods for QR Decomposition

Several algorithms can be used to compute the QR decomposition of a matrix A. The choice of method often depends on the matrix properties (size, sparsity) and the required numerical stability. The three most common methods are:

1. Gram-Schmidt Process

The Gram-Schmidt process works by iteratively orthogonalizing the columns of A to produce the columns of Q. Let the columns of A be $a_1, a_2, ..., a_n$ . The process constructs orthonormal vectors $q_1, q_2, ..., q_n$ that span the same space.

Classical Gram-Schmidt (CGS): For $k = 1, 2, 3, ..., n$ :
1. Orthogonalize:

u_k = a_k - proj_{spanq_1,....q_{k-1}} (a_k)

= a_k - \sum_{j=1}^{k-1} (q_j^T a_k) q_j

2. Normalize:

q_k = \frac{u_k}{||u_k||_2}

The elements of R are given by $r_{jk} = q_j^T a_k$ for $j < k$ , and $r_{kk} = ||u_k||_2$ .

Modified Gram-Schmidt (MGS): This variant improves numerical stability by changing the order of operations. It orthogonalizes vectors against the already computed q vectors at each step. While mathematically equivalent in exact arithmetic, MGS typically performs better in floating-point arithmetic by reducing the loss of orthogonality.

Gram-Schmidt provides a constructive proof for QR decomposition but can suffer from numerical instability (loss of orthogonality) in the classical form, especially for ill-conditioned matrices. MGS is generally preferred for stability.

2. Householder Reflections

This method uses a sequence of Householder reflections (or transformations) to zero out the elements below the main diagonal of A, transforming it into R. A Householder reflection reflects a vector across a hyperplane. The reflection matrix H is orthogonal and symmetric ( $H = H^T, H^TH = I$ ).

For an m x n matrix A, the process involves applying m-1 or n reflections (depending on the shape). In step k, a reflection H_k is constructed to zero out the sub-diagonal elements of the k-th column without affecting the previous columns' upper triangular structure.

H_k = I - 2 \frac{v_k v_k^T}{v_k^T v_k}

Where $v_k$ is the Householder vector chosen appropriately. The sequence of transformations looks like: $H_n \ ... \ H_2 \ H_1 \ A = R$ .

Then, Q is formed by the product of these reflections: Q = H₁^T H₂^T ... H_n^T = H₁ H₂ ... H_n

Q = H_1^T \ H_2^T \ ... \ H_n^T

Q = H_1 \ H_2 \ ... \ H_n

Since H_k is symmetric.

Householder reflections are generally considered the most numerically stable and efficient method for dense matrices of moderate to large size.

3. Givens Rotations

Givens rotations are planar rotations used to selectively zero out specific off-diagonal elements of the matrix. A Givens rotation matrix G(i, j, θ) affects only the i-th and j-th rows (or columns) and corresponds to a rotation by angle θ in the (i, j) coordinate plane.

G(i, j, \theta) = \begin{bmatrix} \ddots & & & & & \\ & c & & s & & \\ & & \ddots & & & \\ & -s & & c & & \\ & & & & \ddots \end{bmatrix} \quad (c=\cos\theta, s=\sin\theta)

By applying a sequence of Givens rotations, one can systematically eliminate all sub-diagonal elements, transforming A into R. The product of the transposes of these Givens rotation matrices forms Q.

G_k \ ... \ G_2 \ G_1 \ A = R

Q = G_1^T \ G_2^T \ ... \ G_k^T

Givens rotations are particularly useful for sparse matrices because they can target specific elements without affecting others as much as Householder reflections. They are also advantageous in parallel computing environments. However, they typically require more floating-point operations than Householder for dense matrices.

QR Decomposition for Rectangular Matrices

QR decomposition is not limited to square matrices; it is equally applicable and widely used for rectangular matrices (m x n). As mentioned earlier, the decomposition can be expressed in two main forms: full QR and reduced (thin) QR.

Full QR Decomposition (m x n matrix A)

In the full QR decomposition, A = QR, Q is always a square orthogonal matrix of size m x m, and R is an m x n matrix with the same shape as A but upper trapezoidal.

A = Q R = \underbrace{\begin{bmatrix} q_{1} & \dots & q_{n} & q_{n+1} & \dots & q_{m} \end{bmatrix}}_{Q: m \times m} \underbrace{\begin{bmatrix} R_{11} \\ 0 \end{bmatrix}}_{R: m \times n}

Here, $R_{11}$ is an n x n upper triangular matrix (if m ≥ n), and the block below it is an (m-n) x n zero matrix. If m < n, R is [ $R_{11} \ R_{12}$ ] where R₁₁ is m x m upper triangular. The columns $q_1, q_2, ... q_m$ form an orthonormal basis for the entire space R^m (or C^m). The columns $q_1, q_2, ... q_n$ form an orthonormal basis for the column space of A, Col(A), if A has full column rank.

Reduced (Thin) QR Decomposition (m x n matrix A, m ≥ n)

The reduced QR decomposition is often more practical, especially when m is much larger than n. It only computes the factors necessary to reconstruct A. If A is m x n with m ≥ n, then A = QR where Q is an m x n matrix with orthonormal columns ( $Q^TQ = I_n$ ), and R is an n x n upper triangular matrix.

A = Q R = \underbrace{\begin{bmatrix} q_{1} & \dots & q_{n} \end{bmatrix}}_{Q: m \times n} \underbrace{\begin{bmatrix} r_{11} & \dots & r_{1n} \\ & \ddots & \vdots \\ & & r_{nn} \end{bmatrix}}_{R: n \times n}

This form arises naturally from algorithms like Gram-Schmidt or when using Householder/Givens if only the first n columns of Q are explicitly formed. The columns of this Q still form an orthonormal basis for Col(A) (assuming full column rank). This form is sufficient for solving least-squares problems Ax ≈ b, as $Q^TA = R$ and $Q^Tb$ can be computed efficiently.

Most numerical software libraries provide options for computing either the full or the reduced QR decomposition. The choice depends on the specific application. Our calculator typically provides the form most relevant to common use cases, often the reduced QR when m > n.

What is a QR Decomposition Calculator?

A QR Decomposition Calculator is an online tool or software function that automatically computes the QR factorization of a user-provided matrix A. Instead of performing the steps of Gram-Schmidt, Householder reflections, or Givens rotations manually, the user simply inputs the matrix elements. The calculator then applies an efficient and numerically stable algorithm (often based on Householder reflections for dense matrices) to compute and display the resulting orthogonal matrix Q and upper triangular matrix R such that A = QR.

These calculators are valuable for:

Educational Purposes: Helping students verify their manual calculations and understand the structure of Q and R.
Quick Computation: Providing rapid factorization for engineers and scientists who need the decomposition for subsequent analysis (like solving least squares).
Avoiding Tedium and Errors: Automating the complex and potentially error-prone steps involved in the manual computation.

Essentially, a QR decomposition calculator serves as a convenient interface to powerful numerical linear algebra routines, making this fundamental factorization accessible without needing specialized software or manual effort.

How to Use This QR Decomposition Calculator?

Step 1

Select the size of your matrix (2x2 up to 5x5)

Step 2

Enter the matrix elements in the provided grid

Step 3

Click Calculate to perform QR decomposition

Step 4

View the orthogonal matrix Q and upper triangular matrix R

Step 5

Use AI explanation for detailed understanding

Features of Calxify's QR Decomposition Calculator

Accurate Calculations

Get precise Q and R matrices using the Gram-Schmidt process.

Step-by-Step Solutions

Follow the detailed orthogonalization process with clear explanations.

Matrix Size Flexibility

Handle matrices of various sizes, from 2x2 up to 5x5.

Educational Support

Learn about QR decomposition through comprehensive explanations.

AI-Powered Insights

Get intelligent explanations of calculations and their significance.

Detailed Information About Related Topics

QR decomposition is deeply connected to several other important concepts in linear algebra and numerical analysis. Understanding these connections provides a broader perspective on its utility.

Determinant of a Matrix

For a square n x n matrix A, the QR decomposition A = QR provides a way to compute its determinant. Since det(A) = det(Q)det(R), and for a real orthogonal matrix Q, det(Q) = ±1, we have |det(A)| = |det(R)|. Since R is upper triangular, its determinant is simply the product of its diagonal elements: $det(R) = r_{11} \ r_{22} \ \dots \ r_{nn}$ .

det(A) = (\pm 1) \times \prod_{i=1}^{n} r_{ii}

Therefore, computing the QR decomposition allows for a numerically stable way to find the determinant's magnitude. The sign depends on det(Q), which relates to whether Q involves reflections.

Singular Value Decomposition (SVD)

SVD ( $A = U \sum{V^T}$ ) is another fundamental matrix factorization. While distinct from QR, QR decomposition is often used as an intermediate step in algorithms to compute the SVD, particularly algorithms based on eigenvalue decompositions of $A^TA$ or $AA^T$ , or iterative bidiagonalization methods. Both factorizations reveal information about the matrix's structure, rank, and column/row spaces, but SVD provides orthonormal bases for both the domain and codomain spaces, linked by singular values.

Least Squares Problem

Perhaps the most important application of QR decomposition is solving the linear least-squares problem: find x that minimizes ${||A_x - b||}_2$ , especially when A is rectangular (m > n) and has full column rank. If A = QR (reduced), the problem becomes minimizing ${||QR_x - b||}_2$ . Since Q has orthonormal columns, Q^TQ = I, and multiplying by Q^T (an orthogonal transformation) preserves the Euclidean norm:

||QRx - b||_2 = ||Q^T(QRx - b)||_2 = ||Rx - Q^Tb||_2

The normal equations $A^TA_x = A^Tb$ are equivalent to $R^TQ^TQR_x = R^TQ^Tb$ $R^TQ^TQR_x = R^TQ^Tb$ which simplifies to $R^TR_x = R^TQ^Tb$ . Since R is invertible (if A has full column rank), this further simplifies to $R_x = Q^Tb$ . This is an upper triangular system that can be easily and stably solved for x using back-substitution. This approach avoids forming A^TA, which can worsen the condition number.

Cholesky Decomposition

Cholesky decomposition (A = LL^T or A = R^TR, where L is lower triangular and R is upper triangular) applies only to symmetric (or Hermitian) positive definite matrices. While both QR and Cholesky involve triangular factors, Cholesky is much more restrictive in its applicability but computationally cheaper when applicable. QR can be used to solve least squares via the normal equations: A^TAx = A^Tb. The matrix A^TA is symmetric positive definite (if A has full column rank), so Cholesky decomposition can be applied to A^TA. However, using QR decomposition directly on A is generally more numerically stable.

Matrix Rank

The QR decomposition can reveal the rank of matrix A. If A = QR, then rank(A) = rank(R). Since R is upper triangular (or trapezoidal), the rank is equal to the number of non-zero diagonal entries in R (assuming pivoting strategies like Column Pivoting QR are used to handle rank deficiency robustly). In the standard QR, if A is m x n (m ≥ n) and rank(A) < n, some diagonal entries of R (in the reduced QR) will be zero or very close to zero in floating-point arithmetic.

Orthogonality and Orthonormality

These concepts are central to the Q matrix in QR decomposition. A set of vectors is orthogonal if their dot products are zero for any distinct pair (v_i · v_j = 0 for i ≠ j). The set is orthonormal if it is orthogonal and each vector has a unit norm (||v_i||₂ = 1 for all i).

The matrix Q in A = QR has columns that form an orthonormal set. This means Q^TQ = I. If Q is square (full QR), it's an orthogonal matrix, and its rows also form an orthonormal set, satisfying QQ^T = I. Orthonormal bases are highly desirable in numerical linear algebra because they simplify calculations (e.g., projections) and preserve geometric structure (lengths and angles), leading to better numerical stability. The Gram-Schmidt process is fundamentally an algorithm for constructing an orthonormal basis from an arbitrary basis.

Applications of QR Decomposition

QR decomposition is a versatile tool with applications across numerous scientific and engineering domains:

1. Solving Linear Systems and Least Squares

As detailed above, QR decomposition provides a numerically stable method for solving Ax = b and minimizing ||Ax - b||₂. This is crucial in statistical modeling (linear regression), data fitting, signal processing, and control systems.

2. Eigenvalue Problems (QR Algorithm)

The QR algorithm is one of the most important methods for computing eigenvalues and eigenvectors of a matrix. It involves iteratively applying QR decomposition. Starting with A₀ = A, the algorithm computes A_k+1 = R_kQ_k, where A_k = Q_kR_k is the QR decomposition of A_k. Under certain conditions, A_k converges to an upper triangular (or quasi-triangular) matrix whose diagonal entries are the eigenvalues of A.

3. Basis Construction and Orthogonalization

The columns of Q provide an orthonormal basis for the column space of A. This is useful in many areas, including function approximation, feature extraction in machine learning (though PCA/SVD are often preferred), and numerical methods for differential equations.

4. Preconditioning

In iterative methods for solving large linear systems, preconditioners are used to improve convergence rates. Incomplete QR factorizations can serve as effective preconditioners for certain types of matrices.

5. Rank Determination

QR decomposition with column pivoting (QRCP) is a reliable method for determining the numerical rank of a matrix by examining the diagonal elements of R.

Frequently Asked Questions

Q1. What is QR decomposition (or QR factorization)?

•

QR decomposition is a method of breaking a matrix A into the product of two matrices: an orthogonal matrix Q and an upper triangular matrix R, such that A = QR. It is a fundamental technique in linear algebra used for solving systems of equations, eigenvalue problems, and more.

Q2. What do Q and R represent in the QR decomposition A = QR?

•

In the QR decomposition A = QR, Q is an orthogonal (or unitary) matrix whose columns form an orthonormal basis for the column space of A, and R is an upper triangular matrix containing the coefficients of the projections.

Q3. What are the properties of the Q matrix in QR decomposition?

•

The Q matrix has orthonormal columns, meaning each column has a length of one and is orthogonal (perpendicular) to the others. It satisfies QᵀQ = I, where I is the identity matrix.

Q4. What are the properties of the R matrix in QR decomposition?

•

The R matrix is an upper triangular matrix, meaning all entries below the main diagonal are zero. It contains the projections of the original columns of A onto the orthonormal vectors in Q.

Q5. Why is QR decomposition useful or important?

•

QR decomposition is essential for solving linear systems, especially overdetermined systems, for finding eigenvalues, and for ensuring numerical stability in computations like least squares problems.

Q6. What are the main applications of QR decomposition?

•

QR decomposition is widely used in solving linear least squares problems, eigenvalue computations, numerical analysis, signal processing, and optimization tasks.

Q7. How is QR decomposition used to solve linear least squares problems?

•

To solve an overdetermined system Ax ≈ b, QR decomposition expresses A as QR, and then solves the simpler system Rx = Qᵀb using back substitution, making the problem more stable and efficient.

Q8. How is QR decomposition related to eigenvalues and eigenvectors?

•

QR decomposition is a crucial step in iterative algorithms like the QR algorithm, which is used to compute the eigenvalues (and sometimes eigenvectors) of a matrix.

Q9. What is the QR algorithm for finding eigenvalues?

•

The QR algorithm repeatedly applies QR decomposition and matrix multiplication to a matrix to converge towards a triangular form whose diagonal elements approximate the eigenvalues of the original matrix.

Q10. How do you compute the QR decomposition of a matrix?

•

QR decomposition can be computed using algorithms like the Gram-Schmidt process, Householder reflections, or Givens rotations. Our Calxify QR Decomposition Calculator instantly performs these computations online for you.

Q11. What are the different methods for calculating QR decomposition?

•

The main methods are the classical Gram-Schmidt process, modified Gram-Schmidt process, Householder reflections, and Givens rotations. Householder reflections are the most numerically stable.

Q12. How does the Gram-Schmidt process work for QR decomposition?

•

The Gram-Schmidt process orthonormalizes the columns of the matrix by subtracting projections onto earlier vectors and normalizing, constructing the Q matrix and forming R from the coefficients.

Q13. How do Householder reflections work for QR decomposition?

•

Householder reflections use reflection matrices to zero out subdiagonal elements of A, transforming it into an upper triangular matrix R while building the orthogonal matrix Q efficiently.

Q14. How do Givens rotations work for QR decomposition?

•

Givens rotations use a sequence of rotations to zero out individual elements below the diagonal. They are particularly useful for sparse matrices where most elements are already zero.

Q15. Which method for QR decomposition is the most numerically stable?

•

Householder reflections are considered the most numerically stable method for QR decomposition, especially for large matrices or when high accuracy is required.

Q16. Is the QR decomposition of a matrix unique?

•

QR decomposition is unique if we require that the diagonal entries of R be positive. Without this constraint, multiple valid QR decompositions can exist for a given matrix.

Q17. What is the difference between full QR and reduced (or thin) QR decomposition?

•

Full QR decomposition results in a square Q matrix, while reduced (or thin) QR decomposition produces a smaller Q with only as many columns as necessary to span the column space of A, making it more efficient for rectangular matrices.

Q18. Can QR decomposition be applied to rectangular matrices?

•

Yes, QR decomposition can be performed on any m × n matrix where m ≥ n. It is often used for solving least squares problems and for data fitting when the matrix is not square.

Q19. Does the Q matrix in QR decomposition always have full column rank?

•

Yes, in a reduced QR decomposition, the Q matrix has orthonormal columns and thus full column rank, corresponding to the rank of the original matrix A.

Q20. How is QR decomposition related to the determinant of a matrix?

•

If A is a square matrix and A = QR, then the determinant of A is the product of the diagonal entries of R times the determinant of Q. Since Q is orthogonal, its determinant is ±1.

Q21. What is the difference between QR decomposition and LU decomposition?

•

QR decomposition factors a matrix into an orthogonal matrix and an upper triangular matrix, while LU decomposition factors it into a lower and an upper triangular matrix. QR is generally used for least squares problems, while LU is used for solving linear systems.

Q22. What is the difference between QR decomposition and Singular Value Decomposition (SVD)?

•

QR decomposition expresses A as QR with orthogonal and triangular matrices, while SVD decomposes A into UΣVᵀ, involving orthogonal matrices and a diagonal matrix of singular values. SVD is more powerful and works for any matrix, but is more computationally expensive.

Q23. How does QR decomposition help improve numerical stability compared to normal equations in least squares?

•

Using QR decomposition avoids squaring the condition number of the matrix, which happens with normal equations, thereby greatly improving numerical stability and reducing the risk of round-off errors.

Q24. What is the relationship between QR decomposition and RQ, QL, or LQ decompositions?

•

While QR decomposition factors A as Q times R, RQ decomposition factors A as R times Q. Similarly, LQ and QL decompositions factor A as a product of a lower triangular and an orthogonal matrix but in different orders. They are variations used depending on the application and matrix structure.