This article is about a matrix symmetric about its diagonal. For a matrix symmetric about its center, see Centrosymmetric matrix.
For matrices with symmetry over the complex number field, see Hermitian matrix.
Symmetry of a 5×5 matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if denotes the entry in the th row and th column then
for all indices and
Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative.
In linear algebra, a real symmetric matrix represents a self-adjoint operator[1] represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric matrix refers to one which has real-valued entries. Symmetric matrices appear naturally in a variety of applications, and typical numerical linear algebra software makes special accommodations for them.
^Jesús Rojo García (1986). Álgebra lineal (in Spanish) (2nd ed.). Editorial AC. ISBN 84-7288-120-2.
a symmetricmatrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{ is symmetric}}\iff...
In mathematics, a symmetricmatrix M {\displaystyle M} with real entries is positive-definite if the real number z T M z {\displaystyle z^{\operatorname...
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...
which maps any skew-symmetricmatrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although...
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...
is symmetric. Moreover, a symmetricmatrix A is uniquely determined by the corresponding quadratic form. Under an equivalence C, the symmetricmatrix A...
The symmetrically normalized Laplacian matrix is symmetric if and only if the adjacency matrix is symmetric. For a non-symmetric adjacency matrix of a...
matrices can be factorized in this way. When the matrix being factorized is a normal or real symmetricmatrix, the decomposition is called "spectral decomposition"...
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...
is a fixed 2 n × 2 n {\displaystyle 2n\times 2n} nonsingular, skew-symmetricmatrix. This definition can be extended to 2 n × 2 n {\displaystyle 2n\times...
special orthogonal group), the differential of a rotation is a skew-symmetricmatrix A T = − A {\displaystyle A^{\mathsf {T}}=-A} in the tangent space s...
orthogonal, symmetric, or skew-symmetric, respectively) matrices are normal and therefore possess this property. Comment: For any real symmetricmatrix A, the...
theorem using skew-symmetric matrices and the tensor formulation, the following identities are useful. Let [R] be the skew symmetricmatrix associated with...
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...
mathematics, the determinant of an m×m skew-symmetricmatrix can always be written as the square of a polynomial in the matrix entries, a polynomial with integer...
of a tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form...
{\displaystyle 2\times 2} matrix would be necessary to fully characterize the two-dimensional variation. Any covariance matrix is symmetric and positive semi-definite...
this defines a symmetric bilinear form on V. This is an example of a symmetric bilinear form which is not associated to any symmetricmatrix (since the vector...
dimension bivectors can be identified with skew-symmetric matrices, so the product between a skew-symmetricmatrix and vector is equivalent to the grade-1 part...
G_{ij}=B\left(v_{i},v_{j}\right)} . The matrix will be symmetric if the bilinear form B {\displaystyle B} is symmetric. In Riemannian geometry, given an embedded...
mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetricmatrix J = [ 0 n I n − I n 0 n ] {\displaystyle...
{\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...
equal). Consequently, only square matrices can be symmetric. The entries of a symmetricmatrix are symmetric with respect to the main diagonal. So if the entries...
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...
diagonal matrix refers to square matrices, which can be specified explicitly as a square diagonal matrix. A square diagonal matrix is a symmetricmatrix, so...
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