Digital Signal Processing Reference
In-Depth Information
3.Giventwosetsofnumbers
(positive)
satisfying the multiplicative majorization property (21.59), there exists a
P
{
λ
k
}
(possibly complex) and
{
σ
k
}
×
P
matrix whose eigenvalues are
{
λ
k
}
and singular values are
{
σ
k
}
.
The first property is similar to the property described in Ex. 21.2 for additive
majorization. The last two properties are very similar to the relation between
additive majorization and Hermitian matrices summarized in Sec. 21.5.
21.7 Summary and conclusions
Schur-convex functions and the theory of majorization have been applied increas-
ingly in many problems in digital communications. This chapter has provided
an extensive review of these mathematical tools, to enable the reader to study
some of the advanced literature in this area. In the next few pages we summarize
the key definitions and examples in the form of tables.
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