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  2. Moore–Penrose inverse - Wikipedia

    en.wikipedia.org/wiki/Moore–Penrose_inverse

    The Python package NumPy provides a pseudoinverse calculation through its functions matrix.I and linalg.pinv; its pinv uses the SVD-based algorithm. SciPy adds a function scipy.linalg.pinv that uses a least-squares solver. The MASS package for R provides a calculation of the Moore–Penrose inverse through the ginv function. [24]

  3. Sherman–Morrison formula - Wikipedia

    en.wikipedia.org/wiki/Sherman–Morrison_formula

    A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case +) if and only if = =. We first verify that the right hand side ( Y {\displaystyle Y} ) satisfies X Y = I {\displaystyle XY=I} .

  4. List of named matrices - Wikipedia

    en.wikipedia.org/wiki/List_of_named_matrices

    Matrix entries are given by the divisor function; entires of the inverse are given by the Möbius function. a ij are 1 if i divides j or if j = 1; otherwise, a ij = 0. A (0, 1)-matrix. Shift matrix: A matrix with ones on the superdiagonal or subdiagonal and zeroes elsewhere. a ij = δ i+1,j or a ij = δ i−1,j

  5. Jacobian matrix and determinant - Wikipedia

    en.wikipedia.org/wiki/Jacobian_matrix_and...

    According to the inverse function theorem, the matrix inverse of the Jacobian matrix of an invertible function f : R n → R n is the Jacobian matrix of the inverse function. That is, the Jacobian matrix of the inverse function at a point p is

  6. Generalized inverse - Wikipedia

    en.wikipedia.org/wiki/Generalized_inverse

    In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element x is an element y that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of ...

  7. Neumann series - Wikipedia

    en.wikipedia.org/wiki/Neumann_series

    Using a truncated Neumann series avoids computation of an explicit matrix inverse, which reduces the complexity of linear data detection from cubic to square. [1] Another application is the theory of propagation graphs which takes advantage of Neumann series to derive closed form expressions for transfer functions.

  8. Singular value decomposition - Wikipedia

    en.wikipedia.org/wiki/Singular_value_decomposition

    Specifically, the singular value decomposition of an complex matrix ⁠ ⁠ is a factorization of the form =, where ⁠ ⁠ is an ⁠ ⁠ complex unitary matrix, is an rectangular diagonal matrix with non-negative real numbers on the diagonal, ⁠ ⁠ is an complex unitary matrix, and is the conjugate transpose of ⁠ ⁠. Such decomposition ...

  9. Inverse Gaussian distribution - Wikipedia

    en.wikipedia.org/wiki/Inverse_Gaussian_distribution

    Its cumulant generating function (logarithm of the characteristic function) [contradictory] is the inverse of the cumulant generating function of a Gaussian random variable. To indicate that a random variable X is inverse Gaussian-distributed with mean μ and shape parameter λ we write X ∼ IG ⁡ ( μ , λ ) {\displaystyle X\sim ...