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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.
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 ...
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 ...
If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa. Working in a fixed basis, can be represented by a matrix.
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.
For example, it maps the positive real numbers to the interval [−1, 1]. Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is
Although multiplication by the centering matrix is not a computationally efficient way of removing the mean from a vector, it is a convenient analytical tool. It can be used not only to remove the mean of a single vector, but also of multiple vectors stored in the rows or columns of an m-by-n matrix .