Editing Characteristic polynomial (section)

{{Use American English|date = April 2019}}
{{short description|Polynomial whose roots are the eigenvalues of a matrix}}
{{About|the characteristic polynomial of a matrix or of an endomorphism of vector spaces|the characteristic polynomial of a matroid|Matroid|that of a graded poset|Graded poset}}

In [[linear algebra]], the '''characteristic polynomial''' of a [[square matrix]] is a [[polynomial]] which is [[Invariant (mathematics)|invariant]] under [[matrix similarity]] and has the [[eigenvalues]] as [[Root of a polynomial|roots]]. It has the [[determinant]] and the [[Trace (linear algebra)|trace]] of the matrix among its coefficients. The '''characteristic polynomial''' of an [[endomorphism]] of a finite-dimensional [[vector space]] is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a [[Basis (linear algebra)|basis]]). The '''characteristic equation''', also known as the '''determinantal equation''',<ref>{{cite book |last=Guillemin |first=Ernst |title=Introductory Circuit Theory |author-link=Ernst Guillemin |date=1953 |url=https://archive.org/details/introductorycirc0000guil |publisher=Wiley |pages=366, 541 |isbn=0471330663}}</ref><ref>{{cite journal |last1=Forsythe |first1=George E. |last2=Motzkin |first2=Theodore |date=January 1952 |title=An Extension of Gauss' Transformation for Improving the Condition of Systems of Linear Equations |url=https://www.ams.org/journals/mcom/1952-06-037/S0025-5718-1952-0048162-0/S0025-5718-1952-0048162-0.pdf |journal= Mathematics of Computation|volume=6 |issue=37 |pages=18–34 |doi=10.1090/S0025-5718-1952-0048162-0 |access-date=3 October 2020|doi-access=free }}</ref><ref>{{cite journal |last=Frank |first=Evelyn |date=1946 |title=On the zeros of polynomials with complex coefficients |journal=Bulletin of the American Mathematical Society |volume=52 |issue=2 |pages=144–157 |doi=10.1090/S0002-9904-1946-08526-2 |doi-access=free |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-52/issue-2/On-the-zeros-of-polynomials-with-complex-coefficients/bams/1183507703.pdf }}</ref> is the equation obtained by equating the characteristic polynomial to zero.

In [[spectral graph theory]], the '''characteristic polynomial of a [[Graph (discrete mathematics)|graph]]''' is the characteristic polynomial of its [[adjacency matrix]].<ref>{{cite web
| url = http://mathworld.wolfram.com/CharacteristicPolynomial.html
| title = Characteristic Polynomial of a Graph – Wolfram MathWorld
|access-date = August 26, 2011}}</ref>