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Its eigenfunctions form a basis of the function space on which the operator is defined [5] As a consequence, in many important cases, the eigenfunctions of the Hermitian operator form an orthonormal basis. In these cases, an arbitrary function can be expressed as a linear combination of the eigenfunctions of the Hermitian operator.
These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
The compatibility theorem tells us that a common basis of eigenfunctions of ^ and ^ can be found. Now if each pair of the eigenvalues ( a n , b n ) {\displaystyle (a_{n},b_{n})} uniquely specifies a state vector of this basis, we claim to have formed a CSCO: the set { A , B } {\displaystyle \{A,B\}} .
The covariance function K X satisfies the definition of a Mercer kernel. By Mercer's theorem, there consequently exists a set λ k, e k (t) of eigenvalues and eigenfunctions of T K X forming an orthonormal basis of L 2 ([a,b]), and K X can be expressed as (,) = = ()
This function ω(λ) plays the role of the characteristic polynomial of D. Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of D and that each non-zero eigenspace is one-dimensional.
Using the Leibniz formula for determinants, the left-hand side of equation is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1) n λ n. This polynomial is called the characteristic polynomial of A.
The i th basis function is chosen to be orthogonal to the basis functions from the first through i − 1, and to minimize the residual variance. That is, the basis functions are chosen to be different from each other, and to account for as much variance as possible.
The eigenfunctions of the transfer operator are usually fractals. When the logarithm of the transfer operator corresponds to a quantum Hamiltonian , the eigenvalues will typically be very closely spaced, and thus even a very narrow and carefully selected ensemble of quantum states will encompass a large number of very different fractal ...