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In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by A T (among other notations).
It is important to note, that the ordering R of Hadamard matrix’s rows (against the Sylvester-Hadamard matrix) does not depend on the vector . Has been proven[5] that, if X {\displaystyle X} is unit vector (i.e. ∥ X ∥ = 1 {\displaystyle \parallel X\parallel =1} ), then T r s {\displaystyle Trs} matrix (obtained as it was described above ...
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 ≡ ...
Performing an in-place transpose (in-situ transpose) is most difficult when N ≠ M, i.e. for a non-square (rectangular) matrix, where it involves a complex permutation of the data elements, with many cycles of length greater than 2.
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.
R is a programming language for statistical computing and data visualization. It has been adopted in the fields of data mining, bioinformatics and data analysis. [9] The core R language is augmented by a large number of extension packages, containing reusable code, documentation, and sample data. R software is open-source and free software.
Transposition, producing the transpose of a matrix A T, which is computed by swapping columns for rows in the matrix A; Transpose of a linear map; Transposition (logic), a rule of replacement in philosophical logic; Transpose relation, another name for converse relation
In the language of category theory, taking the dual of vector spaces and the transpose of linear maps is therefore a contravariant functor from the category of vector spaces over to itself. One can identify t ( t u ) {\displaystyle {}^{t}\left({}^{t}u\right)} with u {\displaystyle u} using the natural injection into the double dual.