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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. Woodbury matrix identity - Wikipedia

    en.wikipedia.org/wiki/Woodbury_matrix_identity

    A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.

  4. 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} .

  5. Invertible matrix - Wikipedia

    en.wikipedia.org/wiki/Invertible_matrix

    Although an explicit inverse is not necessary to estimate the vector of unknowns, it is the easiest way to estimate their accuracy and os found in the diagonal of a matrix inverse (the posterior covariance matrix of the vector of unknowns). However, faster algorithms to compute only the diagonal entries of a matrix inverse are known in many cases.

  6. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I]

  7. 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 ...

  8. LU decomposition - Wikipedia

    en.wikipedia.org/wiki/LU_decomposition

    The above example of matrices demonstrates that matrix product of top row and leftmost columns of involved matrices plays special role for to succeed. Let us mark consecutive versions of matrices with (), (), … and then let us write matrix product () = () in such way that these rows and columns are separated from the rest.

  9. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating ...