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Taking the inner product of each side of this equation with an arbitrary basis function u i (t), = () = = (), = =. This is the matrix multiplication Ab = c written in summation notation and is a matrix equivalent of the operator D acting upon the function f ( t ) expressed in the orthonormal basis.
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations =, …, + = +, giving a k-dimensional system of the first order in the stacked variable vector [+] in terms of its once-lagged value, and taking the characteristic equation of this system's matrix.
This basis can be used to determine an "almost diagonal matrix" in Jordan normal form, similar to , which is useful in computing certain matrix functions of . [9] The matrix J {\displaystyle J} is also useful in solving the system of linear differential equations x ′ = A x , {\displaystyle \mathbf {x} '=A\mathbf {x} ,} where A {\displaystyle ...
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
That is, if is a function on the real line and is a self-adjoint operator, we wish to define the operator (). The spectral theorem shows that if T {\displaystyle T} is represented as the operator of multiplication by h {\displaystyle h} , then f ( T ) {\displaystyle f(T)} is the operator of multiplication by the composition f ∘ h ...
We write the eigenvalue equation in position coordinates, ^ = = recalling that ^ simply multiplies the wave-functions by the function , in the position representation. Since the function x {\displaystyle \mathrm {x} } is variable while x 0 {\displaystyle x_{0}} is a constant, ψ {\displaystyle \psi } must be zero everywhere except at the point ...
We call p(λ) the characteristic polynomial, and the equation, called the characteristic equation, is an N th-order polynomial equation in the unknown λ. This equation will have N λ distinct solutions, where 1 ≤ N λ ≤ N. The set of solutions, that is, the eigenvalues, is called the spectrum of A. [1] [2] [3]
The number of these known functions is the size of the basis set. The expansion coefficients are also numbers. The number of known functions included in the expansion, the same as that of coefficients, is the dimension of the Hamiltonian matrix that will be generated. The statement of the theorem follows. [1] [2]