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*, 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* − *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];
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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 26 apr 2021