“Gersgorin Circle Theorem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gersgorin Circle Theorem

释义

Gersgorin Circle Theorem

Gives a region in the Complex Plane containing all the Eigenvalues of a Complex Square Matrix. Let

(1)

and define

(2)

Then each Eigenvalue of the Matrix A of order

is in at least one of the disks

(3)

The theorem can be made stronger as follows. Let

be an Integer with

, then each Eigenvalue ofA is either in one of the disks

(4)

or in one of the regions

(5)

where

is the sum of magnitudes of the

largest off-diagonal elements in column

References

Brualdi, R. A. and Mellendorf, S. ``Regions in the Complex Plane Containing the Eigenvalues of a Matrix.'' Amer. Math. Monthly 101, 975-985, 1994.

Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, pp. 1120-1121, 1979.

Taussky-Todd, O. ``A Recurring Theorem on Determinants.'' Amer. Math. Monthly 56, 672-676, 1949.

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