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If the matrix is symmetric indefinite, it may be still decomposed as = where is a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, and is a direct sum of symmetric and blocks, which is called Bunch–Kaufman decomposition [6]
The inverse of a Lehmer matrix is a tridiagonal matrix, where the superdiagonal and subdiagonal have strictly negative entries. Consider again the n×n A and m×m B Lehmer matrices, where m > n . A rather peculiar property of their inverses is that A −1 is nearly a submatrix of B −1 , except for the A −1 n,n element, which is not equal to ...
The identity matrix commutes with all matrices. Jordan blocks commute with upper triangular matrices that have the same value along bands. If the product of two symmetric matrices is symmetric, then they must commute. That also means that every diagonal matrix commutes with all other diagonal matrices. [9] [10] Circulant matrices commute.
Define the n × n matrix A by = (,). The matrix A is a symmetric matrix exactly due to symmetry of the bilinear form. If we let the n×1 matrix x represent the vector v with respect to this basis, and similarly let the n×1 matrix y represent the vector w, then (,) is given by :
This is the same matrix as defines a Givens rotation, but for Jacobi rotations the choice of angle is different (very roughly half as large), since the rotation is applied on both sides simultaneously. It is not necessary to calculate the angle itself to apply the rotation. Using Kronecker delta notation, the matrix entries can be written:
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.
Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.; The product of two bisymmetric matrices is a centrosymmetric matrix. Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.
Symmetric centrosymmetric matrices are sometimes called bisymmetric matrices. When the ground field is the real numbers, it has been shown that bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix. [3]