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Characterizations and bounds for weighted sums of eigenvalues of normal and Hermitian matrices

Journal Name:

Mathematica Æterna

Publication Year:

2011

Key Words:

Author Name	University of Author
Jorma K. Merikoski	University of Tampere / Tampereen yliopisto
Ravinder Kumar

Abstract (2. Language):

Let A ∈ Cn×n be normal with eigenvalues 1, . . . , n, and let t1, . . . , tn ∈ C. It is well-known that max ∈Sn |t1(1) + · · · + tn(n)| = max n|t1u∗1Au1 + · · · + tnu∗n Aun| {u1, . . . , un} ⊂o Cno. Here Sn denotes the symmetric group of order n, and ⊂o means “is an orthonormal subset of . . . ”. If A is Hermitian and 1 ≥ · · · ≥ n, and if t1, . . . , tn ∈ R satisfy t1 ≥ · · · ≥ tn, then t11 + · · · + tnn = maxnt1u∗1Au1 + · · · + tnu∗n Aun | {u1, . . . , un} ⊂o Cno and tn1 + · · · + t1n = min nt1u∗1Au1 + · · · + tnu∗n Aun | {u1, . . . , un} ⊂o Cno. We present bounds for the left-hand sides of all these equations by suitable choices of u1, . . . , un.

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FULL TEXT (PDF):

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6

329-346

English

REFERENCES

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346 Jorma K. Merikoski and Ravinder Kumar
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