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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). [1] The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. [2]
The last property given above shows that if one views as a linear transformation from Hilbert space to , then the matrix corresponds to the adjoint operator of . The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis.
The map which sends a matrix to its transpose is an involution because the transpose is well defined for any matrix and obeys the law (AB) T = B T A T, which has the same form of interaction with multiplication as taking inverses has in the general linear group (which is a subgroup of the full linear monoid
For a specific basis, any linear operator can be represented by a matrix T. Every matrix has a transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices. Since elementwise complex conjugation is an independent involution, the conjugate transpose or Hermitian adjoint is also an involution.
In linear algebra, the adjugate or classical adjoint of a square matrix A, adj(A), is the transpose of its cofactor matrix. [1] [2] It is occasionally known as adjunct matrix, [3] [4] or "adjoint", [5] though that normally refers to a different concept, the adjoint operator which for a matrix is the conjugate transpose.
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each c i {\displaystyle c_{i}} is a p × p {\displaystyle p\times p} square matrix , then the n p × n p {\displaystyle np\times np} matrix C {\displaystyle C} is called a block-circulant matrix .