Characteristic
Polynomials and Eigenvalues for Matrices |

**In General**

$ __Characteristic polynomial__:
The polynomial one obtains by expanding det(*A*–*λ*I)
in powers of *λ*, as det(*A*–*λ*I)
= ∑_{i} *σ*_{i}
*λ** ^{i}*,

*σ*_{n} =
1, *σ*_{n–1} =
–tr *A*, *σ*_{n–2} =
\({1\over2}\)(tr *A*)^{2} – \({1\over2}\)tr
*A*^{2}, *σ*_{n–3} =
–det *A*, ...;

They are also defined recursively by

*σ*_{i} =
(–1)* ^{n–i}* {

where {*A*}:= tr(*A*), {*A*_{1},
*A*_{2}}:= tr(*A*_{1})
tr(*A*_{2}) – tr(*A*_{1}*A*_{2}),
and in general

{*A*_{1}, *A*_{2},
..., *A*_{n+1}}:=
tr(*A*_{n+1})
{*A*_{1}, *A*_{2},
..., *A*_{n}} – ∑_{i=1}* ^{n}* {

$ __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(*A*–*λ*I) = 0 , or ∑_{i} *σ*_{i} *A ^{i}* =
0 ,

$ __Eigenvalues__:
The solutions of the characteristic equation for a matrix *A*, i.e.,
the values of *λ* such that
det(*A*–*λ*I) = 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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2016