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Every real symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix. If and are real symmetric matrices that commute, then they can be simultaneously diagonalized by an orthogonal matrix: [2] there exists a basis of such that every element of the basis is an eigenvector for both and . Every real symmetric matrix is ...
Any real symplectic matrix can be decomposed as a product of three matrices: = ′, where and ′ are both symplectic and orthogonal, and is positive-definite and diagonal. [6] This decomposition is closely related to the singular value decomposition of a matrix and is known as an 'Euler' or 'Bloch-Messiah' decomposition.
Consequently, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if the entries are written as A = (a ij), then a ij = a ji, for all indices i and j. For example, the following 3×3 matrix is symmetric:
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.
If we use a skew-symmetric matrix, every 3 × 3 skew-symmetric matrix is determined by 3 parameters, and so at first glance, the parameter space is R 3. Exponentiating such a matrix results in an orthogonal 3 × 3 matrix of determinant 1 – in other words, a rotation matrix, but this is a many-to-one map.
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 ...
Any symmetric permutation of the arrowhead matrix, , where P is a permutation matrix, is a (permuted) arrowhead matrix. Real symmetric arrowhead matrices are used in some algorithms for finding of eigenvalues and eigenvectors .
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]