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Unformatted text preview: asis for R n . We will see that
these complications do not occur for symmetric matrices.
Spectral Theorem.1 Suppose that A is a symmetric n × n matrix with real
entries. Then its eigenvalues are real and eigenvectors corresponding to distinct
eigenvectors are orthogonal. Moreover, there is an n × n orthogonal matrix B
of determinant one such that B −1 AB = B T AB is diagonal.
Sketch of proof: The reader may want to skip our sketch of the proof at ﬁrst,
returning after studying some of the examples presented later in this section.
We will assume the following two facts, which are proven rigorously in advanced
courses on mathematical analysis:
1. Any continuous function on a sphere (of arbitrary dimension) assumes its
maximum and minimum values.
2. The points at which the maximum and minimum values are assumed can
be found by the method of Lagrange multipliers (a method usually discussed in vector calculus courses).
1 This is called the “spectral theorem” because the spectrum is another name for the set of
eigenvalues of a matrix. 30 The equation of the sphere S n−1 in R n is x...
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 Winter '14
 Equations

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