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  2. Eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/Eigenvalue_algorithm

    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 ...

  3. Eigenvalues and eigenvectors - Wikipedia

    en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

    Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes.

  4. Eigenfunction - Wikipedia

    en.wikipedia.org/wiki/Eigenfunction

    This solution of the vibrating drum problem is, at any point in time, an eigenfunction of the Laplace operator on a disk.. In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

  5. QR algorithm - Wikipedia

    en.wikipedia.org/wiki/QR_algorithm

    The eigenvalues of a matrix are always computable. We will now discuss how these difficulties manifest in the basic QR algorithm. This is illustrated in Figure 2. Recall that the ellipses represent positive-definite symmetric matrices. As the two eigenvalues of the input matrix approach each other, the input ellipse changes into a circle.

  6. Rayleigh–Ritz method - Wikipedia

    en.wikipedia.org/wiki/Rayleigh–Ritz_method

    Truncated singular value decomposition (SVD) in numerical linear algebra can also use the Rayleigh–Ritz method to find approximations to left and right singular vectors of the matrix of size in given subspaces by turning the singular value problem into an eigenvalue problem.

  7. Eigenvalues and eigenvectors of the second derivative

    en.wikipedia.org/wiki/Eigenvalues_and...

    Note that there are 2n + 1 of these values, but only the first n + 1 are unique. The (n + 1)th value gives us the zero vector as an eigenvector with eigenvalue 0, which is trivial. This can be seen by returning to the original recurrence. So we consider only the first n of these values to be the n eigenvalues of the Dirichlet - Neumann problem.

  8. Jacobi eigenvalue algorithm - Wikipedia

    en.wikipedia.org/wiki/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 ...

  9. Quadratic eigenvalue problem - Wikipedia

    en.wikipedia.org/wiki/Quadratic_eigenvalue_problem

    Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing: ″ + ′ + = Where (), and ,,.If all quadratic eigenvalues of () = + + are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as