QR Decomposition Calculator – Matrix Factorization
Decompose a matrix into Q (orthogonal) and R (upper triangular) matrices
Table of Contents
How to Use
- Set the matrix dimensions (rows and columns)
- Enter the values for each element of the matrix
- Click calculate to perform QR decomposition
- View the resulting Q and R matrices
What is QR Decomposition?
QR decomposition (also called QR factorization) is a way of expressing a matrix A as a product of two matrices: Q and R. Matrix Q is an orthogonal matrix (its columns are orthonormal vectors), and R is an upper triangular matrix.
The decomposition is written as A = QR, where Q has orthonormal columns (Q^T Q = I) and R has zeros below its main diagonal.
The Gram-Schmidt Process
This calculator uses the Gram-Schmidt orthogonalization process to compute the QR decomposition. The process works by:
- Taking each column of A in sequence
- Subtracting projections onto previously computed orthonormal vectors
- Normalizing the result to get a unit vector
- Recording the projection coefficients in matrix R
Properties of Q and R
Matrix Q (Orthogonal):
- Columns are orthonormal (perpendicular unit vectors)
- Q^T Q = I (identity matrix)
- Preserves vector lengths and angles
Matrix R (Upper Triangular):
- All entries below the main diagonal are zero
- Diagonal entries are the norms of the orthogonalized vectors
- Off-diagonal entries are projection coefficients
Applications of QR Decomposition
- Solving linear least squares problems
- Computing eigenvalues (QR algorithm)
- Solving systems of linear equations
- Signal processing and data compression
- Machine learning algorithms
Frequently Asked Questions
- What is the difference between QR and LU decomposition?
- QR decomposition produces an orthogonal matrix Q and upper triangular R, while LU decomposition produces a lower triangular L and upper triangular U. QR is more numerically stable and is preferred for least squares problems.
- Can any matrix be QR decomposed?
- Any real matrix with linearly independent columns can be QR decomposed. For matrices with linearly dependent columns, a modified version called QR with column pivoting can be used.
- What does it mean for Q to be orthogonal?
- An orthogonal matrix Q has the property that Q^T Q = I (identity matrix). This means its columns are mutually perpendicular unit vectors, and multiplying by Q preserves lengths and angles.
- How is QR decomposition used in least squares?
- For the least squares problem Ax ≈ b, QR decomposition transforms it to Rx = Q^T b, which is easy to solve by back substitution since R is upper triangular.