Qrdecomposition Calculator
Our free fractions calculator solves qrdecomposition problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
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]]
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy