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In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa.
The matrix representation of the quaternion product is convenient for programming quaternion computations using matrix algebra, which is true for dual quaternion operations as well. The quaternion product AC is a linear transformation by the operator A of the components of the quaternion C, therefore there is a matrix representation of A ...
In matrix inversion however, instead of vector b, we have matrix B, where B is an n-by-p matrix, so that we are trying to find a matrix X (also a n-by-p matrix): A X = L U X = B . {\displaystyle AX=LUX=B.}
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 ...
The matrix vectorization operation can be written in terms of a linear sum. Let X be an m × n matrix that we want to vectorize, and let e i be the i -th canonical basis vector for the n -dimensional space, that is e i = [ 0 , … , 0 , 1 , 0 , … , 0 ] T {\textstyle \mathbf {e} _{i}=\left[0,\dots ,0,1,0,\dots ,0\right]^{\mathrm {T} }} .
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.
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U can be written as U = e iH, where e indicates the matrix exponential, i is the imaginary unit, and H is a Hermitian matrix. For any nonnegative integer n, the set of all n × n unitary matrices with matrix multiplication forms a group, called the unitary group U(n). Every square matrix with unit Euclidean norm is the average of two unitary ...