Lecture 10 Summary

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Operation count for Gram-Schmidt. Discussed loss of orthogonality in classical Gram-Schmidt, using a simple example, especially in the case where the matrix has nearly dependent columns to begin with. Showed modified Gram-Schmidt and argued how it (mostly) fixes the problem.

Introduced Householder QR, emphasized the inherent stability properties of multiplying by a sequence of unitary matrices (as shown in pset 2). Showed how we can convert a matrix to upper-triangular form (superficially similar to Gaussian elimination) via unitary Householder reflectors.