Class 14: Array Factorizations for Square Matrices

Review from previous class

Sweep operations

Sweep operations done using the augmented X'X matrix are fast and popular, supporting such tools as stepwise regression. They are, however, less accurate numerically that altertives such as that offered by the QR decomposition.

As an aside, if you return to our examples of the sweep function, you will see a good example for why one wants object-oriented programming. Look at all of the pieces that are lying around, unconnected.

LU Factorization methods

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

The remainder of the examples illustrate the use of the Cholesky, QR, eigen and singular value decompositions.

Status of Projects

• Hopefully Assignment #3 has you moving along.
• My page remains near the lead...

Square Matrix Factorizations

LU decomposition

This is a basic building block, most often used indirectly (via inverses, for example) rather than directly. Once you see how it gets computed, though, the others also make more sense.

Cholesky factorization

The most common matrix square root, this is basically an LU decomposition in which the factors are equal, but transposed. The statistical application is to generating correlated normal samples, as in regression or time series.

Spectral representation (eigenvectors and values)

All square matrices can be represented through a nice cannonical form using eigenvectors and eigenvalues. The spectral representation leads to a different version of the square root of a matrix. The version in LispStat is limited to symmetric matrices.

Lisp script for today's class

class14.lsp

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