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matrix is symmetric matrix. T r {\displaystyle Tr} matrix is persymmetric matrix , i.e. it is symmetric with respect to the northeast-to-southwest diagonal too. Every one row and column of T r {\displaystyle Tr} matrix consists all n elements of given vector X {\displaystyle X} without repetition.
Download QR code; Print/export ... a symmetric matrix is a square matrix that is equal to its transpose. Formally, ... The transpose of a symmetrizable matrix is ...
If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n. Furthermore, these products are symmetric matrices. Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
MATLAB version 6.0 and later; SciPy, and thus SciPy-relied software FuncDesigner, SageMath, PythonXY; It appears as a built-in routine (for lu, backslash, and forward slash) in MATLAB, and includes a MATLAB interface, a C-callable interface, and a Fortran-callable interface. Note that "UMFPACK" is pronounced in two syllables, "Umph Pack".
The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b
Suppose that the 2n-by-2n matrix A is written as the block matrix = [] where a, b, c, and d are n-by-n matrices. Then the condition that A be Hamiltonian is equivalent to requiring that the matrices b and c are symmetric, and that a + d T = 0.
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. For such matrices, the half-vectorization is sometimes more useful than the ...
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...