--- title: QR Factorization description: QR factorization decomposes a matrix into an orthogonal matrix and an upper triangular matrix. Learners will understand how to compute this decomposition and apply it to solve least squares and eigenvalue problems. category: mathematics subcategory: numerical-linear-algebra difficulty: beginner, intermediate, advanced url: /subject/qr-factorization --- # QR Factorization QR factorization decomposes a matrix into an orthogonal matrix and an upper triangular matrix. Learners will understand how to compute this decomposition and apply it to solve least squares and eigenvalue problems. ## Available Resources 1 Videos • 1 Books • 2 Courses • 4 Websites ## Courses ### 1. Linear Algebra (MIT 18.06) **Author:** Gilbert Strang Matrix theory and linear algebra: systems of equations, elimination, vector spaces and subspaces, orthogonality and least squares, determinants, eigenvalues and positive definite matrices. Includes 35 lecture videos plus problem sets and exams with solutions, giving the fluency needed for applied mathematics, engineering and data science. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ **Tags:** linear-algebra, vector-spaces, eigenvalues, least-squares, matrix-factorization ### 2. Computational Science and Engineering I (MIT 18.085) **Author:** Gilbert Strang Applied linear algebra for networks, structures and estimation, followed by equilibrium differential equations, Laplace's equation, boundary-value problems, calculus of variations, Fourier series and the discrete Fourier transform. Includes 50 lecture videos, problem sets and exams with solutions, and programming assignments. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-fall-2008/ **Tags:** applied-linear-algebra, boundary-value-problems, calculus-of-variations, fourier-analysis, differential-equations ## Books ### 1. Accuracy and Stability of Numerical Algorithms, 2nd Edition **Author:** Nicholas J. Higham The terminal rung: once a learner can compute QR three ways, this is where the question of which one is actually trustworthy, and by how much, gets a proven answer. Rounding-error analysis reference whose QR chapter proves backward stability of Householder and Givens triangularization, quantifies loss of orthogonality in classical versus modified Gram-Schmidt, and treats column pivoting plus least-squares perturbation bounds. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0898715210?tag=edmonddante07-20 **Tags:** numerical-linear-algebra, qr-factorization, rounding-error-analysis, backward-stability, householder, gram-schmidt ## Websites ### 1. Numerical Computing with MATLAB, Chapter 5: Least Squares **Author:** Cleve Moler Free chapter applying orthogonal factorization to overdetermined systems: curve fitting and design matrices, why normal equations lose accuracy, Householder QR, pseudoinverse and rank deficiency, illustrated with runnable MATLAB code and US census data. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.mathworks.com/content/dam/mathworks/mathworks-dot-com/moler/leastsquares.pdf **Tags:** least-squares, qr-factorization, householder, pseudoinverse, matlab, numerical-linear-algebra ### 2. ALAFF Week 3: The QR Decomposition **Author:** Robert van de Geijn, Margaret Myers Full chapter of a free graduate text covering classical and modified Gram-Schmidt, the catastrophic cancellation that degrades their orthogonality, and Householder QR derived in partitioned algorithmic form with flop counts, homework, and MATLAB implementations. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.cs.utexas.edu/~flame/laff/alaff/chapter03-qr.html **Tags:** qr-factorization, gram-schmidt, householder, numerical-linear-algebra, numerical-stability ### 3. What Is a QR Factorization? **Author:** Nicholas J. Higham Compact reference article defining full and reduced QR, stating existence and uniqueness conditions, listing the three standard computation routes (Householder reflectors, Givens rotations, Gram-Schmidt), and connecting the factorization to least squares and column pivoting. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://nhigham.com/2020/11/10/what-is-a-qr-factorization/ **Tags:** qr-factorization, householder, givens-rotations, gram-schmidt, least-squares, numerical-linear-algebra ### 4. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Videos ### 1. Lecture 17: Orthogonal Matrices and Gram-Schmidt (MIT 18.06SC) **Author:** Gilbert Strang Explains orthonormal bases and why QᵀQ = I simplifies projections and least squares, then works through the Gram-Schmidt procedure step by step to obtain A = QR. Includes the lecture video, summary notes, a recitation video and problems with solutions for practicing the factorization. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/pages/least-squares-determinants-and-eigenvalues/orthogonal-matrices-and-gram-schmidt/ **Tags:** gram-schmidt, qr-decomposition, orthogonal-matrices, orthonormal-bases, linear-algebra --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/qr-factorization