Characteristic polynomial

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.

This works because the diagonal entries are also the eigenvalues of this matrix.

Another example uses hyperbolic functions of a hyperbolic angle φ. For the matrix take

Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.

This proof only applies to matrices and polynomials over complex numbers (or any algebraically closed field). In that case, the characteristic polynomial of any square matrix can be always factorized as

The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations (on a time scale of a century, that is, slow compared to annual motion) of planetary orbits, according to Lagrange's theory of oscillations.