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computed with ﬂoating-point arithmetic can vary with the algorithm used and is
often diﬀerent than rank computed with exact arithmetic (recall Exercise 2.2.4). Copyright c 2000 SIAM Buy online from SIAM
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592
Chapter 7
Eigenvalues and Eigenvectors
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Please report violations to [email protected] Furthermore, computing higher-index eigenvalues with ﬂoating-point arithmetic
is fraught with peril. To see why, consider the matrix
⎛
⎞
01
.. ..
⎜
⎟
.
.
⎜
⎟
L( ) = ⎜
whose characteristic equation is λn − = 0.
.. 1 ⎟
⎝
⎠
.
0 n×n
For = 0, zero is the only eigenvalue (and it has index n ), but for all > 0,
there are n distinct eigenvalues given by 1/n e2kπi/n for k = 0, 1, . . . , n − 1. For
example, if n = 32, and if
changes from 0 to 10−16 , then the eigenvalues
of L( ) change in magnitude from 0 to 10−1/2 ≈ .316, which is substantial for
such a small perturbation. Sensitivities of this kind present signiﬁcant problems
for ﬂoating-point algorithms. In addition to showing that high-index eigenvalues
are sensitive to small perturbations, this example also shows that the Jordan
structure is highly discontinuous. L(0) is in Jordan form, and there is just one
Jordan block of size n, but for all = 0, the Jordan form of L( ) is a diagonal
matrix—i.e., there are n Jordan blocks of size 1 × 1. Lest you think that this
example somehow is an isolated case, recall from Example 7.3.6 (p. 532) that
every matrix in C n×n is arbitrarily close to a diagonalizable matrix.
All of the above observations make it clear that it’s hard to have faith in
a Jordan form that has been computed with ﬂoating-point arithmetic. Consequently, numerical computation of Jordan forms is generally avoided. D
E T
H IG
R Example 7.8.2 Y
P The Jordan form of A conveys complete information about the eigenva...

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