1. What is QR Decomposition?

QR decomposition factors a matrix A into a product of an orthogonal matrix Q and an upper triangular matrix R. This decomposition is fundamental in numerical linear algebra and has applications in solving linear systems, least squares problems, and eigenvalue computations.

2. How Does Householder QR Work?

The Householder method uses a series of orthogonal transformations (reflectors) to zero out subdiagonal elements:

Where:

• Each \( Q_i \) is a Householder reflector matrix
• The product \( Q = Q_1^TQ_2^T...Q_n^T \) is orthogonal
• R is upper triangular

Explanation: Householder transformations are numerically stable and more accurate for ill-conditioned matrices than Gram-Schmidt.

3. Importance of QR Decomposition

Details: QR decomposition is essential for solving least squares problems, computing eigenvalues (via QR algorithm), and is more numerically stable than Gaussian elimination for ill-conditioned matrices.

4. Using the Calculator

Tips: Enter matrix elements separated by commas for columns and semicolons for rows. The matrix must be rectangular (all rows same length) with numeric values.

5. Frequently Asked Questions (FAQ)

Q1: Why use Householder instead of Gram-Schmidt?
A: Householder is numerically stable and preserves orthogonality better than classical Gram-Schmidt.

Q2: What matrices can be QR decomposed?
A: Any real or complex square or rectangular matrix has a QR decomposition.

Q3: Is the decomposition unique?
A: The decomposition is unique if A is full rank and we require positive diagonal elements in R.

Q4: What's the computational complexity?
A: For an m×n matrix, Householder QR requires O(2mn² - 2n³/3) floating point operations.

Q5: How is this different from LU decomposition?
A: QR uses orthogonal transformations (stable for any matrix), while LU uses Gaussian elimination (can be unstable for ill-conditioned matrices).