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The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix . In this formulation, the defining equation is. where is a scalar and is a matrix. Any row vector satisfying this equation is called a left eigenvector of and is its associated eigenvalue.
hide. In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called ...
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 ...
Although undeniably a cornerstone of organic chemistry, Hückel's concepts were undeservedly unrecognized for two decades. Pauling and Wheland characterized his approach as "cumbersome" at the time, and their competing resonance theory was relatively easier to understand for chemists without fundamental physics background, even if they couldn't ...
Rayleigh–Ritz method. The Rayleigh–Ritz method is a direct numerical method of approximating eigenvalues, originated in the context of solving physical boundary value problems and named after Lord Rayleigh and Walther Ritz. In this method, an infinite-dimensional linear operator is approximated by a finite-dimensional compression, on which ...
The Lanczos algorithm is most often brought up in the context of finding the eigenvalues and eigenvectors of a matrix, but whereas an ordinary diagonalization of a matrix would make eigenvectors and eigenvalues apparent from inspection, the same is not true for the tridiagonalization performed by the Lanczos algorithm; nontrivial additional steps are needed to compute even a single eigenvalue ...
Jacobi eigenvalue algorithm. In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix (a process known as diagonalization). It is named after Carl Gustav Jacob Jacobi, who first proposed the method in 1846, [1] but only became widely ...
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.