1. What is QR Decomposition?

QR decomposition is a matrix factorization technique that decomposes a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. It's widely used in linear algebra for solving linear systems, least squares problems, and eigenvalue computations.

2. How Does QR Decomposition Work?

The decomposition is typically computed using:

Where:

• \( Q \) — Orthogonal matrix (\( Q^TQ = I \))
• \( R \) — Upper triangular matrix

Methods: Common algorithms include Gram-Schmidt process, Householder reflections, and Givens rotations.

3. Applications of QR Decomposition

Details: QR decomposition is fundamental in numerical linear algebra, used for solving least squares problems, computing eigenvalues (QR algorithm), and matrix inversion.

4. Using the Calculator

Tips: Enter your matrix as space or comma separated values, with rows separated by spaces. For example, "1,2,3 4,5,6" represents a 2×3 matrix.

5. Frequently Asked Questions (FAQ)

Q1: What makes Q orthogonal?
A: Q is orthogonal if its columns are orthonormal vectors (mutually perpendicular unit vectors).

Q2: Is QR decomposition unique?
A: The decomposition is unique if A is invertible and we require positive diagonal elements in R.

Q3: How is this different from LU decomposition?
A: QR works for any matrix (even rectangular), while LU requires square matrices and works with lower/upper triangular matrices.

Q4: What's the computational complexity?
A: For an n×n matrix, it's O(n³) operations, similar to other decomposition methods.

Q5: When does QR decomposition fail?
A: It doesn't fail for any matrix, but numerical stability can be an issue for ill-conditioned matrices.