**Unformatted text preview: **possess complex eigenvalues and eigenvectors. O
C As we have seen, computing eigenvalues boils down to solving a polynomial
equation. But determining solutions to polynomial equations can be a formidable
task. It was proven in the nineteenth century that it’s impossible to express
the roots of a general polynomial of degree ﬁve or higher using radicals of the
coeﬃcients. This means that there does not exist a generalized version of the
quadratic formula for polynomials of degree greater than four, and general polynomial equations cannot be solved by a ﬁnite number of arithmetic operations
n
involving +,−,×,÷, √ . Unlike solving Ax = b, the eigenvalue problem generally requires an inﬁnite algorithm, so all practical eigenvalue computations are
accomplished by iterative methods—some are discussed later. Copyright c 2000 SIAM Buy online from SIAM
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494
Chapter 7
Eigenvalues and Eigenvectors
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For theoretical work, and for textbook-type problems, it’s helpful to express
the characteristic equation in terms of the principal minors. Recall that an r × r
principal submatrix of An×n is a submatrix that lies on the same set of r
rows and columns, and an r × r principal minor is the determinant of an r × r
principal submatrix. In other words, r × r principal minors are obtained by
deleting the same set of n−r rows and columns, and there are n = n!/r!(n−r)!
r
such minors. For example, the 1 × 1 principal minors of
⎛
⎞
−3
1 −3
A = ⎝ 20
3 10 ⎠
(7.1.4)
2 −2
4 D
E are the diagonal entries −3, 3, and 4. The 2 × 2 principal minors are
−3
20 1
= −29,
3 −3
2 −3
= −6,
4 and 3
−2 10
= 32,
4 T
H and the only 3 × 3 principal minor is det (A) = −18.
Related to the principal minors are the symmetric functions of the eigenvalues. The k th symmetric function of λ1 , λ2 , . . . , λn is deﬁned to be the sum
of the product of the eigenvalues taken k at a time. T...

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