a symmetric matrix is a square matrix that is equal to its transpose. Formally, A  is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff...

 skew-symmetric ⟺ A T = − A . {\displaystyle A{\text{ skew-symmetric}}\quad \iff \quad A^{\textsf {T}}=-A.} In terms of the entries of the matrix, if a...

In mathematics, a symmetric matrix M {\displaystyle M} with real entries is positive-definite if the real number x ⊤ M x {\displaystyle \mathbf {x} ^{\top...

thus pi j = pj i, and the product matrix (pi j) is symmetric. Similarly, the product AT A is a symmetric matrix. A quick proof of the symmetry of A...

A^{*}A=AA^{*}} . If a real square matrix is symmetric, skew-symmetric, or orthogonal, then it is normal. If a complex square matrix is Hermitian, skew-Hermitian...

is symmetric. Moreover, a symmetric matrix A is uniquely determined by the corresponding quadratic form. Under an equivalence C, the symmetric matrix A...

be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A {\displaystyle A} is denoted by A H , {\displaystyle...

identity is also a permutation matrix. A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well...

which maps any skew-symmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although...

special orthogonal group), the differential of a rotation is a skew-symmetric matrix A T = − A {\displaystyle A^{\mathsf {T}}=-A} in the tangent space s...

= AT, is a symmetric matrix. If instead, A is equal to the negative of its transpose, that is, A = −AT, then A is a skew-symmetric matrix. In complex...

is a fixed 2 n × 2 n {\displaystyle 2n\times 2n} nonsingular, skew-symmetric matrix. This definition can be extended to 2 n × 2 n {\displaystyle 2n\times...

The symmetrically normalized Laplacian matrix is symmetric if and only if the adjacency matrix is symmetric. For a non-symmetric adjacency matrix of a...

{\displaystyle 2\times 2} matrix would be necessary to fully characterize the two-dimensional variation. Any covariance matrix is symmetric and positive semi-definite...

1s on the diagonal. Real symmetric matrices are diagonalizable by orthogonal matrices; i.e., given a real symmetric matrix A {\displaystyle A} , Q T...

issue doesn't arise when A is real and symmetric, resulting in a simple algorithm: % Given a real symmetric 3x3 matrix A, compute the eigenvalues % Note that...

{\displaystyle {\frac {1}{a_{0}}}A} . The inverse of a nonsingular symmetric Toeplitz matrix has the representation A − 1 = 1 α 0 ( B B T − C C T ) {\displaystyle...

of a tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form...

theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful. Let [R] be the skew symmetric matrix associated with...

mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix J = [ 0 n I n − I n 0 n ] {\displaystyle...

moments can be described by a symmetric 3-by-3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around...

Correlation matrix — a symmetric n×n matrix, formed by the pairwise correlation coefficients of several random variables. Covariance matrix — a symmetric n×n...

matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetric matrix, the decomposition is called "spectral decomposition"...

the determinant of an m-by-m skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer...

} A further saving is possible when the matrix is symmetric. For example, consider a symmetric 6-by-6 matrix with an upper bandwidth of 2: [ A 11 A 12...

matrices can be understood as the complex versions of real skew-symmetric matrices, or as the matrix analogue of the purely imaginary numbers. The set of all...

orthogonal, symmetric, or skew-symmetric, respectively) matrices are normal and therefore possess this property. Comment: For any real symmetric matrix A, the...

this defines a symmetric bilinear form on V. This is an example of a symmetric bilinear form which is not associated to any symmetric matrix (since the vector...

equal). 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...

which implies that Q is a symmetric matrix. If q is congruent to 3 mod 4 then −1 is not a square, and Q is a skew-symmetric matrix. When q is a prime number...