Qrdecomposition Calculator
Our free fractions calculator solves qrdecomposition problems. Get worked examples, visual aids, and downloadable results.
Formula
A = QR
Where Q is an orthogonal matrix (Q^T Q = I) with orthonormal columns computed via the Gram-Schmidt process, and R is an upper triangular matrix containing the projection coefficients and norms.
Worked Examples
Example 1: QR Decomposition of a 3x2 Matrix
Problem: Find the QR decomposition of A = [[1,1],[0,1],[1,0]].
Solution: Column 1: a1 = [1,0,1], ||a1|| = sqrt(2)\nq1 = [1/sqrt(2), 0, 1/sqrt(2)]\nr11 = sqrt(2) = 1.4142\n\nColumn 2: a2 = [1,1,0]\nr12 = q1 . a2 = 1/sqrt(2) = 0.7071\nu2 = a2 - r12*q1 = [0.5, 1, -0.5]\nr22 = ||u2|| = sqrt(1.5) = 1.2247\nq2 = u2/r22 = [0.4082, 0.8165, -0.4082]
Result: Q = [[0.7071, 0.4082], [0, 0.8165], [0.7071, -0.4082]], R = [[1.4142, 0.7071], [0, 1.2247]]
Example 2: QR of a Square 2x2 System
Problem: Find QR of [[3,1],[4,2]] to solve 3x+y=5, 4x+2y=6.
Solution: a1 = [3,4], ||a1|| = 5, q1 = [0.6, 0.8]\nr11 = 5, r12 = q1.[1,2] = 0.6+1.6 = 2.2\nu2 = [1,2] - 2.2*[0.6,0.8] = [-0.32, 0.24]\nr22 = 0.4, q2 = [-0.8, 0.6]\nSolve Rx = Q^T b by back substitution
Result: R = [[5, 2.2], [0, 0.4]], Q = [[0.6, -0.8], [0.8, 0.6]]
Frequently Asked Questions
How accurate are the results from Qrdecomposition Calculator?
All calculations use established mathematical formulas and are performed with high-precision arithmetic. Results are accurate to the precision shown. For critical decisions in finance, medicine, or engineering, always verify results with a qualified professional.
Is Qrdecomposition Calculator free to use?
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Enter values as precisely as possible using the correct units for each field. Check that you have selected the right unit (e.g. kilograms vs pounds, meters vs feet) before calculating. Rounding inputs early can reduce output precision.
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