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Concept
Characteristic polynomial
Summary
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the 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 base (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero.
In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
In linear algebra, eigenvalues and eigenvectors play a fundamental role, since, given a linear transformation, an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector.
More precisely, if the transformation is represented by a square matrix an eigenvector and the corresponding eigenvalue must satisfy the equation
or, equivalently,
where is the identity matrix, and
(although the zero vector satisfies this equation for every it is not considered an eigenvector).
It follows that the matrix must be singular, and its determinant
must be zero.
In other words, the eigenvalues of A are the roots of
which is a monic polynomial in x of degree n if A is a n×n matrix. This polynomial is the characteristic polynomial of A.
Consider an matrix The characteristic polynomial of denoted by is the polynomial defined by
where denotes the identity matrix.
Some authors define the characteristic polynomial to be That polynomial differs from the one defined here by a sign so it makes no difference for properties like having as roots the eigenvalues of ; however the definition above always gives a monic polynomial, whereas the alternative definition is monic only when is even.
Official source
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