This emphasizes the way in which the reflection should be computed. But I want to define it in a different way, by a MATLAB anonymous function. It is usual to define a Householder reflection by a matrix. QR = Q*RĪ Householder reflection is characterized by a vector $u$, which, following Pete's convention, is normalized to have Let's check that $Q$ and $R$ reproduce $A$. Operations based upon orthogonal matrices are very desirable in numeric computation because they do not magnify errors, either those inherited from the underlying data, or those introduced by floating point arithmetic.Ī nifty example, taken from the Wikipedia page on "QR decomposition", is unusual because $A$, $R$, and a renormalization of $Q$ all have integer entries. But, more importantly, the decomposition is part of the fundamental MATLAB linear equation solver denoted by backslash, " \ ", as well as both the eig and svd functions for dense matrices. The decomposition is available explicitly from the MATLAB function qr. The letter Q is a substitute for the letter O from "orthogonal" and the letter R is from "right", an alternative for "upper". The QR decomposition expresses a matrix as the product of an orthogonal matrix and an upper triangular matrix. The reference for my blog today is his book "Matrix Algorithms, Volume I: Basic Decompositions", published by SIAM. Pete has written several books on numerical linear algebra. It somehow goes back through his father to his grandfather and maybe great grandfather, who were also nicknamed "Pete". He has never been able to satisfactorily explain the origins of "Pete" to me. Stewart is a Distinguished University Professor Emeritus at the Department of Computer Science, University of Maryland. I wrote about them in MATLAB News & Notes. The Gatlinburg Conferences are now called the Householder Conferences. A photo of the 1964 committee is available in your MATLAB demos directory. He introduced what he called elementary Hermitian matrices in a paper in the Journal of the ACM in 1958.Īlston was head of the organizing committee for the Gatlinburg Conferences on Numerical Algebra. He was a member of the mathematics division of Oak Ridge National Laboratory for over 20 years, from 1946 until 1969, and was also a Professor at the University of Tennessee.Īlston served as President of both SIAM and ACM. This example uses norm(x)/sqrt(n) to obtain the root-mean-square ( RMS) value of an n-element vector x.Alston Householder (1904-1993) is one of the pioneers of modern numerical linear algebra. On the other hand, MATLAB uses "length" to denote the number of elements n in a vector.
Note that norm(x) is the Euclidean length of a vector x. The Frobenius-norm of matrix A, sqrt(sum(diag(A'* A))). T he infinity norm, or largest row sum of A, max(sum(abs(A'))). The largest singular value (s ame as norm(A)). The 1-norm, or largest column sum of A, max(sum(abs(A)).
Returns a different kind of norm, depending on the value of p. Returns t he largest singular value of A, max(svd(A)). The norm function calculates several different types of matrix norms: The norm of a matrix is a scalar that gives some measure of the magnitude of the elements of the matrix. Norm (MATLAB Functions) MATLAB Function Reference