What is QR-Decomposition?
QR-decomposition, also known as QR factorization, is a fundamental matrix decomposition technique in linear algebra. It decomposes a matrix into two constituent matrices, Q (an orthogonal matrix) and R (an upper triangular matrix). QR-decomposition is used to solve linear systems of equations, to compute eigenvalues and eigenvectors of a matrix, and to solve least squares problems.
Mathematical Definition of QR-Decomposition
Given a matrix A of size m×n with m ≥ n, the QR-decomposition of A is a pair of matrices Q and R such that:
- Q is an m×m orthogonal matrix, meaning that its columns are orthogonal unit vectors (QTQ = I, where I is the identity matrix).
- R is an m×n upper triangular matrix, which means that all the entries below the main diagonal are zero.
- The product of Q and R gives the original matrix A (A = QR).
When A is square (m = n), R is also a square upper triangular matrix, and the factorization is unique if A is invertible and the diagonal elements of R are required to be positive.
Methods for Computing QR-Decomposition
There are several numerical methods to compute the QR-decomposition of a matrix:
- Gram-Schmidt process: This method orthogonalizes the columns of A by subtracting the projection of each column onto the subspace formed by the previous columns. It is conceptually simple but can be numerically unstable.
- Householder reflections: This approach uses a series of reflection matrices to transform A into an upper triangular matrix. It is more numerically stable than the Gram-Schmidt process and is widely used in numerical computing.
- Givens rotations: Givens rotations use a series of rotational matrices to zero out the sub-diagonal entries of A. This method is particularly useful for sparse matrices.
- Modified Gram-Schmidt process: This is a numerically stable version of the Gram-Schmidt process that re-orthogonalizes the vectors to combat the loss of orthogonality due to rounding errors.
Applications of QR-Decomposition
QR-decomposition is used in various applications across scientific computing, engineering, and applied mathematics:
- Solving Linear Systems: QR-decomposition can be used to solve the linear system Ax = b by first decomposing A into QR and then solving Rx = QTb, which is simpler as R is an upper triangular matrix.
- Least Squares Problems: In regression analysis, QR-decomposition is used to solve least squares problems where the goal is to find an approximate solution that minimizes the sum of the squares of the residuals.
- Eigenvalue Computations: The QR algorithm, which iteratively applies QR-decomposition, is a common method for finding the eigenvalues and eigenvectors of a matrix.
- Stability Analysis: In control theory, QR-decomposition is used to analyze the stability of linear systems.
Advantages and Limitations
QR-decomposition is favored for its numerical stability and the orthogonality of matrix Q, which preserves the norm during transformations. However, it can be computationally expensive, especially for large matrices, and there are cases where other decompositions, such as LU-decomposition or singular value decomposition (SVD), might be more appropriate.
Conclusion
QR-decomposition is a versatile and powerful tool in numerical linear algebra. Its ability to break down a matrix into orthogonal and triangular components makes it invaluable for solving a wide range of problems. While there are more efficient methods for certain types of matrices, the QR-decomposition remains a fundamental technique in the toolkit of engineers, scientists, and mathematicians.