Class 16: Discussion of Array Factorizations

Review from previous class

Factorization methods for rectangular matrices

The QR and SV decompositions generalize the Cholesky and spectral decompositions to apply to retangular matrices X, such as the design matrix of predictors in regression.

LU Factorization,S = LU

This factorization is a basic building block, supporting inverse, determinant and solve operations.

Cholesky, S = LL'

This factorization specializes the LU to p.s.d. matrices, leading to a way of finding the "square root" of a matrix. Its also tied closely to regression.

Spectral, S = E D E'

This orthogonal decomposition leads to principal components and an alternative definition of a matrix square root.

QR decomposition

This is a Gram-Schmidt decomposition of a matrix, and leads to a more stable method for solving ill-conditioned systems of equations, such as those in regression with high collinearity.

Singular value decomposition, SVD

The SVD generalizes the notion of eigenvectors. The SVD leads to a very different way to do regression, known as total least squares (TLS). TLS is one approach to the problem of "errors in variables" in regression.

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Review

Lisp script for today's class

• Square matrices (Class 14)
• Rectangular matrices (Class 15)

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