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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 ...
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.
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]
The transformation matrix A = [] preserves the direction of purple vectors parallel to v λ=1 = [1 −1] T and blue vectors parallel to v λ=3 = [1 1] T. The red vectors are not parallel to either eigenvector, so, their directions are changed by the transformation.
The determinant of the matrix equals the product of its eigenvalues. Similarly, the trace of the matrix equals the sum of its eigenvalues. [4] [5] [6] From this point of view, we can define the pseudo-determinant for a singular matrix to be the product of its nonzero eigenvalues (the density of multivariate normal distribution will need this ...
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.
Since eigenvectors are defined up to multiplication by constant, the choice of can be arbitrary in theory; practical aspects of the choice of are discussed below. At every iteration, the vector b k {\displaystyle b_{k}} is multiplied by the matrix ( A − μ I ) − 1 {\displaystyle (A-\mu I)^{-1}} and normalized.
The function is usually required to be a holomorphic function of (in some domain). In general, M ( λ ) {\displaystyle M(\lambda )} could be a linear map , but most commonly it is a finite-dimensional, usually square, matrix.