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The author explores Householder transforms and their applications in signal processing. He shows that these transforms can be viewed as mirror-image reflections of a data vector about any desired hyperplane. The virtue of reflections is that they are covariance invariant, that is, they preserve the covariance matrix of the data. One can construct a finite sequence of such reflections that maps a block of data vectors into a lower rectangular matrix. If only the covariance eigenvalues need to be preserved, one can map into a bidiagonal matrix. The former sparse form is useful for inverting covariance matrices and the latter is useful in finding eigenvalues of covariance matrices.< >