Characteristic Polynomials and Eigenvalues for Matrices  

In General
$ Characteristic polynomial: The polynomial one obtains by expanding det(AλI) in powers of λ, as det(AλI) = ∑i σi λi, i = 0, ..., n; The coefficients σi are invariants of A, and are obtained from traces of powers of A,

σn = 1,   σn–1 = –tr A,   σn–2 = \({1\over2}\)(tr A)2 – \({1\over2}\)tr A2,   σn–3 = –det A,   ...;

They are also defined recursively by

σi = (–1)n–i {A, A, ..., A}/(ni)!   (n i times) ,

where {A}:= tr(A), {A1, A2}:= tr(A1) tr(A2) – tr(A1A2), and in general

{A1, A2, ..., An+1}:= tr(An+1) {A1, A2, ..., An} – ∑i=1n {A1, A2, ..., Ai An+1, ..., An} .

$ Characteristic equation: For an n × n matrix A, the equation one gets by setting the characteristic polynomial to zero (gives the eigenvalues of the matrix as solutions) or substituting A for λ (the equation still holds, the coefficients being invariants),

det(AλI) = 0 ,   or   ∑i σi Ai = 0 ,   i = 0, ..., n .

$ Eigenvalues: The solutions of the characteristic equation for a matrix A, i.e., the values of λ such that det(AλI) = 0.

Special Matrices and Related Topics
@ Eigenvalues: Diaconis BAMS(03) [large unitary matrices, patterns]; > s.a. Frobenius Theorem; operations on matrices [diagonalization].
@ From groups: Bump & Gamburd CMP(06)mp/05 [classical groups, ratios and products].
@ Random matrices: Brézin & Hikami CMP(00), CMP(01); Hughes et al CMP(01).
@ Random matrices, eigenvalues: Krivelevich & Vu mp/00; Witte et al Nonlin(00)mp, mp/00; Semerjian & Cugliandolo JPA(02) [sparse]; Farmer a0709 [unitary].

References
@ General: in Gliozzi & Virasoro NPB(80); Rodrigues JMP(98)mp [trace formulas]; Tapia JPA(07)mp [higher-rank matrices]; Hatzinikitas a0711 [diagrammatic approach to determine coefficients].
> Online resources: see MathWorld page; Wikipedia page.


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