Characteristic Polynomials and Eigenvalues for Matrices |
In General
$ Characteristic polynomial:
The polynomial one obtains by expanding det(A−λI)
in powers of λ, as det(A−λI)
= ∑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}/(n−i)! (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, it is the equation one gets by setting the characteristic polynomial to zero (which gives the eigenvalues of the matrix as solutions) or substituting A for λ (the equation still holds, the coefficients being invariants),
det(A − λI) = 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(A − λI) = 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];
Denton et al BAMS(21)-a1908 [eigenvectors from eigenvalues].
@ Algorithms: Krakoff et al a2104 [QUBO algorithm, symmetric matrices].
> Online resources:
see MathWorld page;
Wikipedia page.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 apr 2021