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  2. Analytic function of a matrix - Wikipedia

    en.wikipedia.org/wiki/Analytic_function_of_a_matrix

    In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of the same size. This is used for defining the exponential of a matrix , which is involved in the closed-form solution of systems of linear differential equations .

  3. Matrix (mathematics) - Wikipedia

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

    For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.

  4. Matrix analysis - Wikipedia

    en.wikipedia.org/wiki/Matrix_analysis

    In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. [1] Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and ...

  5. Matrix calculus - Wikipedia

    en.wikipedia.org/wiki/Matrix_calculus

    In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.

  6. Transformation matrix - Wikipedia

    en.wikipedia.org/wiki/Transformation_matrix

    In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. 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 ...

  7. Matrix exponential - Wikipedia

    en.wikipedia.org/wiki/Matrix_exponential

    For matrix-matrix exponentials, there is a distinction between the left exponential Y X and the right exponential X Y, because the multiplication operator for matrix-to-matrix is not commutative. Moreover, If X is normal and non-singular, then X Y and Y X have the same set of eigenvalues. If X is normal and non-singular, Y is normal, and XY ...

  8. Determinant - Wikipedia

    en.wikipedia.org/wiki/Determinant

    A matrix with entries in a field is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by

  9. 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 ...