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Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an n × m matrix in-place, with O(1) additional storage or at most storage much less than mn. For n ≠ m, this involves a complicated permutation of the data elements that is non-trivial to implement in-place.
A transposition is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation g from above can be written as g = (1 2)(2 5)(3 4).
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
Transposition (mathematics), a permutation which exchanges two elements and keeps all others fixed; 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
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.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry (the complex conjugate of + being , for real numbers and ).
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