Characteristic
Polynomials and Eigenvalues for Matrices| Characteristic
Polynomials and Eigenvalues for Matrices |
In General
$ Characteristic polynomial:
The polynomial one obtains by expanding det(A–
I)
in powers of , as
det(A–I)
= 
i 
i
i, i =
0, ..., n;
The coefficients s are invariants of A, and are obtained from
traces of powers
of A,
n =
1, n–1 =
–tr A, n–2 =
(tr A)2 – tr
A2, n–3 =
–det A, ...;
They are also defined recursively by
i =
(–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,
the equation one gets by setting the characteristic polynomial to zero (gives
the eigenvalues
of the
matrix as solutions) or substituting A for (the
equation still holds,
the coefficients being invariants),
i i Ai =
0, i = 0, ..., n .
$ Eigenvalues:
The solutions of the characteristic equation for a matrix A, i.e.,
the values of such
that det(A–I)
= 0.
Special Matrices and Related Topics
@ Eigenvalues: Diaconis BAMS(03) [large unitary matrices, patterns].
@ From groups: Bump & Gamburd 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].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
8 nov 2007