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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]
There is also an active R community around the tidyverse. For example, there is the TidyTuesday social data project organised by the Data Science Learning Community (DSLC), [ 16 ] where varied real-world datasets are released each week for the community to participate, share, practice, and make learning to work with data easier. [ 17 ]
Vectorization is a unitary transformation from the space of n×n matrices with the Frobenius (or Hilbert–Schmidt) inner product to C n 2: (†) = † (), where the superscript † denotes the conjugate transpose.
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map.
A square matrix is called a projection matrix if it is equal to its square, i.e. if =. [2]: p. 38 A square matrix is called an orthogonal projection matrix if = = for a real matrix, and respectively = = for a complex matrix, where denotes the transpose of and denotes the adjoint or Hermitian transpose of .
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each is a square matrix, then the matrix is called a block-circulant matrix.. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of .
Visual understanding of multiplication by the transpose of a matrix. If A is an orthogonal matrix and B is its transpose, the ij-th element of the product AA T will vanish if i≠j, because the i-th row of A is orthogonal to the j-th row of A.
where is the transpose of . One obtains an isomorphic group (indeed, a conjugate subgroup of GL(p + q)) by replacing g with any symmetric matrix with p positive eigenvalues and q negative ones. Diagonalizing this matrix gives a conjugation of this group with the standard group O(p, q).