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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.
−λ here is the eigenvalue for both differential operators, and T(t) and X(x) are corresponding eigenfunctions. We will now show that solutions for X(x) for values of λ ≤ 0 cannot occur: Suppose that λ < 0. Then there exist real numbers B, C such that
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\}} .
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.
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 (,) = = ()
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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.