# 0 0 8 34 applying the elementary row operations to

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Unformatted text preview: A more complete proof of the theorem is presented in more advanced courses in linear algebra.2 In any case, the method for ﬁnding the orthogonal matrix B such that B T AB is diagonal is relatively simple, at least when the eigenvalues are distinct. Simply let B be the matrix whose columns are unit-length eigenvectors for A. In the case of repeated roots, we must be careful to choose a basis of unit-length eigenvectors for each eigenspace which are perpendicular to each other. Example. The matrix 5 A = 4 0 4 5 0 0 0 1 is symmetric, so its eigenvalues must be real. Its characteristic equation is 0= 5−λ 4 0 4 5−λ 0 0 0 1−λ = [(λ − 5)2 − 16](λ − 1) = (λ2 − 10λ + 9)(λ − 1) = (λ − 1)2 (λ − 9), which has the roots λ1 = 1 with multiplicity two and λ2 = 9 with multiplicity one. Thus we are in the notorious “repeated root” case, which might be expected to cause problems if A were not symmetric. However, since A is symmetric, the Spectral Theorem guarantees that we can ﬁnd a basis for R 3 consisting...
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