QR QR Decomposition Calculator

QR Decomposition Calculator

Compute A = QR via Gram-Schmidt orthogonalization. Q is orthogonal (Qᵀ Q = I), R is upper triangular. Verification included.

QR

QR Decomposition

Gram-Schmidt method: A = QR

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Matrix A (must have ≥ rows than cols for full QR)
Quick Examples

QR Decomposition

Any real matrix A with linearly independent columns can be factored as A = QR where Q is orthogonal and R is upper triangular with positive diagonal entries.

Applications

  • Solving least-squares problems (Ax = b, over-determined systems)
  • Eigenvalue algorithms (QR algorithm)
  • Signal processing and data compression
  • Numerical stability in solving linear systems
$$A = QR, \quad Q^TQ = I, \quad R \text{ upper triangular}$$