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  2. Matrix norm - Wikipedia

    en.wikipedia.org/wiki/Matrix_norm

    Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.

  3. Conjugate gradient method - Wikipedia

    en.wikipedia.org/wiki/Conjugate_gradient_method

    Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite.

  4. Jordan normal form - Wikipedia

    en.wikipedia.org/wiki/Jordan_normal_form

    Sets of representatives of matrix conjugacy classes for Jordan normal form or rational canonical forms in general do not constitute linear or affine subspaces in the ambient matrix spaces. Vladimir Arnold posed [ 16 ] a problem: Find a canonical form of matrices over a field for which the set of representatives of matrix conjugacy classes is a ...

  5. Logarithmic norm - Wikipedia

    en.wikipedia.org/wiki/Logarithmic_norm

    The logarithmic norm was independently introduced by Germund Dahlquist [1] and Sergei Lozinskiĭ in 1958, for square matrices. It has since been extended to nonlinear operators and unbounded operators as well. [2] The logarithmic norm has a wide range of applications, in particular in matrix theory, differential equations and numerical analysis ...

  6. Smith normal form - Wikipedia

    en.wikipedia.org/wiki/Smith_normal_form

    In mathematics, the Smith normal form (sometimes abbreviated SNF [1]) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square ...

  7. Matrix regularization - Wikipedia

    en.wikipedia.org/wiki/Matrix_regularization

    There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix.

  8. Matrix normal distribution - Wikipedia

    en.wikipedia.org/wiki/Matrix_normal_distribution

    The probability density function for the random matrix X (n × p) that follows the matrix normal distribution , (,,) has the form: (,,) = ⁡ ([() ()]) / | | / | | /where denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in , i.e.: the measure corresponding to integration ...

  9. Frobenius normal form - Wikipedia

    en.wikipedia.org/wiki/Frobenius_normal_form

    For each invariant factor f i one takes its companion matrix C f i, and the block diagonal matrix formed from these blocks yields the rational canonical form of A. When the minimal polynomial is identical to the characteristic polynomial (the case k = 1), the Frobenius normal form is the companion matrix of the characteristic polynomial.