Search results
Results from the WOW.Com Content Network
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 ...
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or / / (sometimes called the normalized Laplacian), where is a diagonal matrix with ...
The algebraic connectivity (also known as Fiedler value or Fiedler eigenvalue after Miroslav Fiedler) of a graph G is the second-smallest eigenvalue (counting multiple eigenvalues separately) of the Laplacian matrix of G. [1] This eigenvalue is greater than 0 if and only if G is a connected graph. This is a corollary to the fact that the number ...
In matrix theory, Sylvester's formula or Sylvester's matrix theorem (named after J. J. Sylvester) or Lagrange−Sylvester interpolation expresses an analytic function f(A) of a matrix A as a polynomial in A, in terms of the eigenvalues and eigenvectors of A. [1] [2] It states that [3]
for k := 1 to n−1 do m := k for l := k+1 to n do if e l > e m then m := l endif endfor if k ≠ m then swap e m,e k swap E m,E k endif endfor. 4. The algorithm is written using matrix notation (1 based arrays instead of 0 based). 5. When implementing the algorithm, the part specified using matrix notation must be performed simultaneously. 6.
Hence M = [m 1, m 2] and K = [k 1, k 2]. A mode shape is assumed for the system, with two terms, one of which is weighted by a factor B , e.g. Y = [1, 1] + B [1, −1]. Simple harmonic motion theory says that the velocity at the time when deflection is zero, is the angular frequency ω {\displaystyle \omega } times the deflection (y) at time of ...
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 eigendecomposition (or spectral decomposition) of a diagonalizable matrix is a decomposition of a diagonalizable matrix into a specific canonical form whereby the matrix is represented in terms of its eigenvalues and eigenvectors. The spectral radius of a square matrix is the largest absolute value of its eigenvalues.