Understanding QR Factorization
QR factorization rewrites a matrix A as the product of Q and R. Q contains orthonormal column directions. R contains the upper triangular weights needed to rebuild the original matrix. This split is useful because orthogonal directions are stable during numerical work.
Why This Calculator Helps
Hand work can become slow after only a few rows. This calculator accepts rectangular matrices, decimal values, fractions, and scientific notation. It then applies a Gram-Schmidt based process and reports Q, R, the reconstructed matrix, residual error, rank estimate, and orthogonality check. These outputs help you see both the answer and the quality of the answer.
Practical Uses
QR decomposition appears in least squares fitting, linear regression, eigenvalue routines, signal processing, and numerical solvers. It is often preferred when normal equations would amplify roundoff error. A tall matrix can represent many observations with fewer variables. QR helps solve the closest fitting coefficient vector without forming A transpose A directly.
Rectangular Matrix Support
Square matrices are not required. You may enter tall or wide data sets. The tool builds a thin decomposition and flags weak columns. This is helpful when a problem has repeated columns, noisy readings, or more equations than unknowns.
Reading the Result
The Q matrix should have columns with length one when those columns are independent. Different columns should have dot products near zero. The R matrix stores the scale and projection values. Multiplying Q by R should return the entered matrix. The residual norm tells how close that reconstruction is.
Method Notes
Modified Gram-Schmidt is a practical classroom method. It improves numerical behavior by subtracting projections step by step. Classical Gram-Schmidt is also included for comparison. Both can struggle with nearly dependent columns, so the tolerance field matters. A larger tolerance marks weak columns as dependent sooner. A smaller tolerance keeps more columns active, but may show more rounding noise.
Best Practice
Start with exact rows and reasonable precision. Use six to eight decimals for study. Use higher precision when comparing errors. Check the condition note when columns look similar. Export the CSV for spreadsheets. Save the PDF summary for assignments, reports, or review notes. Keep original row order, because projections depend on the entered column layout shown.