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  2. Norm (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Norm_(mathematics)

    The concept of unit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a square oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square.

  3. 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.

  4. Dual norm - Wikipedia

    en.wikipedia.org/wiki/Dual_norm

    The Frobenius norm defined by ‖ ‖ = = = | | = ⁡ = = {,} is self-dual, i.e., its dual norm is ‖ ‖ ′ = ‖ ‖.. The spectral norm, a special case of the induced norm when =, is defined by the maximum singular values of a matrix, that is, ‖ ‖ = (), has the nuclear norm as its dual norm, which is defined by ‖ ‖ ′ = (), for any matrix where () denote the singular values ...

  5. Law (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Law_(mathematics)

    Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean l 2 {\displaystyle l_{2}} norm , called the canonical or induced norm , where the norm of a vector u {\displaystyle \mathbf {u} } is denoted and defined by

  6. Central tendency - Wikipedia

    en.wikipedia.org/wiki/Central_tendency

    The 2-norm and ∞-norm are strictly convex, and thus (by convex optimization) the minimizer is unique (if it exists), and exists for bounded distributions. Thus standard deviation about the mean is lower than standard deviation about any other point, and the maximum deviation about the midrange is lower than the maximum deviation about any ...

  7. Operator norm - Wikipedia

    en.wikipedia.org/wiki/Operator_norm

    In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces .

  8. Cholesky decomposition - Wikipedia

    en.wikipedia.org/wiki/Cholesky_decomposition

    In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.

  9. Uniform norm - Wikipedia

    en.wikipedia.org/wiki/Uniform_norm

    For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2. In mathematical analysis, the uniform norm (or sup norm) assigns, to real-or complex-valued bounded functions ⁠ ⁠ defined on a set ⁠ ⁠, the non-negative number