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  2. Determinant - Wikipedia

    en.wikipedia.org/wiki/Determinant

    The determinant of a matrix A is commonly denoted det(A), det A, or | A |. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism.

  3. Leibniz formula for determinants - Wikipedia

    en.wikipedia.org/wiki/Leibniz_formula_for...

    (In practical applications of numerical linear algebra, however, explicit computation of the determinant is rarely required.) See, for example, Trefethen & Bau (1997) . The determinant can also be evaluated in fewer than O ( n 3 ) {\displaystyle O(n^{3})} operations by reducing the problem to matrix multiplication , but most such algorithms are ...

  4. Square root of a 2 by 2 matrix - Wikipedia

    en.wikipedia.org/wiki/Square_root_of_a_2_by_2_matrix

    A square root of a 2×2 matrix M is another 2×2 matrix R such that M = R 2, where R 2 stands for the matrix product of R with itself. In general, there can be zero, two, four, or even an infinitude of square-root matrices. In many cases, such a matrix R can be obtained by an explicit formula.

  5. Jacobian matrix and determinant - Wikipedia

    en.wikipedia.org/.../Jacobian_matrix_and_determinant

    When this matrix is square, that is, when the function takes the same number of variables as input as the number of vector components of its output, its determinant is referred to as the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian in literature. [4]

  6. Matrix determinant lemma - Wikipedia

    en.wikipedia.org/wiki/Matrix_determinant_lemma

    The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + v T u). So we have the result:

  7. Second partial derivative test - Wikipedia

    en.wikipedia.org/wiki/Second_partial_derivative_test

    For the general case of an arbitrary number n of variables, there are n sign conditions on the n principal minors of the Hessian matrix that together are equivalent to positive or negative definiteness of the Hessian (Sylvester's criterion): for a local minimum, all the principal minors need to be positive, while for a local maximum, the minors ...

  8. ‘Busiest Thanksgiving ever’ for air travel with more than 18M ...

    www.aol.com/busiest-thanksgiving-ever-air-travel...

    This Thanksgiving is expected to be the busiest ever for air travel, with a record-setting 18 million Americans expected to jet off for their holiday celebrations in the coming days.

  9. Jacobi's formula - Wikipedia

    en.wikipedia.org/wiki/Jacobi's_formula

    In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. [1]If A is a differentiable map from the real numbers to n × n matrices, then