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In example if , and , are two arbitrary selected elements from the same column q of matrix, then, matrix consists one fours of elements (,,,,,), for which are satisfied the equations , =, and , =,. This property, named “Tr-property” is specific to T r {\displaystyle Tr} matrices.
For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by ...
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
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.
In mathematics, especially in linear algebra and matrix theory, the commutation matrix is used for transforming the vectorized form of a matrix into the vectorized form of its transpose. Specifically, the commutation matrix K (m,n) is the nm × mn permutation matrix which, for any m × n matrix A, transforms vec(A) into vec(A T): K (m,n) vec(A ...
For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm (e.g. Frigo & Johnson, 2005), transposing the matrix in memory (to make the columns contiguous) may ...
For example, for the 2×2 matrix = [], the vectorization is = []. The connection between the vectorization of A and the vectorization of its transpose is given by the commutation matrix . Compatibility with Kronecker products
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.