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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).
The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an matrix made up of complex numbers. For an explanation of the notation used here, we begin by representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is,
The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices. In particular, = for matrices A, B, and C of dimensions k×l, l×m, and m×n.
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.
matrix is symmetric matrix.; matrix is persymmetric matrix, i.e. it is symmetric with respect to the northeast-to-southwest diagonal too.; Every one row and column of matrix consists all n elements of given vector without repetition.
On a computer, one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid data movement.
The column space of a matrix A is the set of all linear combinations of the columns in A. If A = [a 1 ⋯ a n], then colsp(A) = span({a 1, ..., a n}). Given a matrix A, the action of the matrix A on a vector x returns a linear combination of the columns of A with the coordinates of x as coefficients; that is, the columns of the matrix generate ...
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 .