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  2. Invertible matrix - Wikipedia

    en.wikipedia.org/wiki/Invertible_matrix

    Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. [ 2 ] Over a field , a square matrix that is not invertible is called singular or degenerate .

  3. Woodbury matrix identity - Wikipedia

    en.wikipedia.org/wiki/Woodbury_matrix_identity

    This can be derived using blockwise matrix inversion. While the identity is primarily used on matrices, it holds in a general ring or in an Ab-category. The Woodbury matrix identity allows cheap computation of inverses and solutions to linear equations. However, little is known about the numerical stability of the formula.

  4. Computational complexity of mathematical operations - Wikipedia

    en.wikipedia.org/wiki/Computational_complexity...

    The complexity of an elementary function is equivalent to that of its inverse, since all elementary functions are analytic and hence invertible by means of Newton's method. In particular, if either exp {\displaystyle \exp } or log {\displaystyle \log } in the complex domain can be computed with some complexity, then that complexity is ...

  5. 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} .

  6. Cholesky decomposition - Wikipedia

    en.wikipedia.org/wiki/Cholesky_decomposition

    The explicit inverse of a Hermitian matrix can be computed by Cholesky decomposition, in a manner similar to solving linear systems, using operations (multiplications). [10] The entire inversion can even be efficiently performed in-place.

  7. Involutory matrix - Wikipedia

    en.wikipedia.org/wiki/Involutory_matrix

    That is, multiplication by the matrix is an involution if and only if =, where is the identity matrix. Involutory matrices are all square roots of the identity matrix. This is a consequence of the fact that any invertible matrix multiplied by its inverse is the identity.

  8. Orthogonal matrix - Wikipedia

    en.wikipedia.org/wiki/Orthogonal_matrix

    The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all n × n orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension ⁠ n(n − 1) / 2 ⁠, called the orthogonal group and denoted by O(n).

  9. Identity matrix - Wikipedia

    en.wikipedia.org/wiki/Identity_matrix

    The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that: