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There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices , or the language of linear ...
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 ...
Functions can be written as a linear combination of the basis functions, = = (), for example through a Fourier expansion of f(t). The coefficients b j can be stacked into an n by 1 column vector b = [b 1 b 2 … b n] T. In some special cases, such as the coefficients of the Fourier series of a sinusoidal function, this column vector has finite ...
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
The Jordan normal form is the most convenient for computation of the matrix functions (though it may be not the best choice for computer computations). Let f(z) be an analytical function of a complex argument. Applying the function on a n×n Jordan block J with eigenvalue λ results in an upper triangular 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 ...
The lowest value among (i=1,2,..,N), , will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i can be obtained by finding the coefficients { c j } {\displaystyle \left\lbrace c_{j}\right\rbrace } from ...
Proof that a common eigenbasis implies commutation. Let {| } be a set of orthonormal states (i.e., | =,) that form a complete eigenbasis for each of the two compatible observables and represented by the self-adjoint operators ^ and ^ with corresponding (real-valued) eigenvalues {} and {}, respectively.