Introduction to Scientific Computing: Orthogonal and Hermitian Matrices in Linear Least Squares
This document discusses key concepts in scientific computing, particularly focusing on orthogonal and Hermitian matrices as used in linear least squares. It explains that a matrix Q is orthogonal if its inverse equals its transpose, highlighting the properties of orthonormal rows and columns. The Householder transformation, essential for zeroing subdiagonal elements of a matrix, is explored, emphasizing efficiency in computation. Additionally, it covers QR decomposition, showcasing how Householder transformations can simplify the solution of overdetermined systems.
Introduction to Scientific Computing: Orthogonal and Hermitian Matrices in Linear Least Squares
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Presentation Transcript
Linear Least Squares Paul Heckbert Computer Science Department Carnegie Mellon University 15-859B - Introduction to Scientific Computing
Orthogonal and Hermitian Matrix A square matrix Q is orthogonal iff Q-1=QT • QTQ = QQT = I • its rows are orthonormal its columns are orthonormal note: orthogonal matrices are often named Q generalization: a matrix is Hermitian iff Q-1=QH where superscript H denotes complex conjugate transpose 15-859B - Introduction to Scientific Computing
Householder Transformations The Householder transformation determined by vector v is: To apply it to a vector x, compute: outer product, nn matrix inner product, scalar scalar 15-859B - Introduction to Scientific Computing
Householder Geometry • Hx is x reflected through the hyperplane perpendicular to v (p : pTv=0) 15-859B - Introduction to Scientific Computing
Householder Properties • H is symmetric, since • H is orthogonal, since 15-859B - Introduction to Scientific Computing
Householder to Zero Matrix Elements We’ll use Householder transformations to zero subdiagonal elements of a matrix. Given any vector a, find the v that determines an H such that, Now solve for v: 15-859B - Introduction to Scientific Computing
Choosing the Vector v 15-859B - Introduction to Scientific Computing
Applying Householder Transforms • Don’t compute Hx explicitly, that costs 3n2 flops. • Instead use the formula given previously, which costs 4n flops (if you pre-compute vTv or pre-normalize vTv=2). • Typically, when using Householder transformations, you never compute the matrix H; it’s only used in derivation and analysis. 15-859B - Introduction to Scientific Computing
QR Decomposition • Householder transformationsare a good way to zero out subdiagonal elements of a matrix. • A is decomposed: • where QT=Hn…H2H1 is the orthogonal (prove!) product of Householders and R is upper triangular. • Overdetermined system Ax=b is transformed into the easy-to-solve 15-859B - Introduction to Scientific Computing
