symmetric matrix

[sə′me·trik ′mā·triks]

(mathematics)

A matrix which equals its transpose.

The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Symmetric Matrix

a square matrix S = ||s_ik|| in which any two elements that are symmetrically located with respect to the principal diagonal are equal: s_ik = s_ki, where i, k = 1, 2,…, n. A symmetric matrix is often treated as the matrix of the coefficients of some quadratic form. The theory of symmetric matrices and the theory of quadratic forms are closely related.

The properties of the spectrum of a symmetric matrix with real elements include the following: (1) all the roots λ₁, λ₂,…, λ_n of the characteristic equation of the matrix are real; and (2) to these roots there correspond n pairwise orthogonal eigenvectors of the matrix, where n is the order of the matrix. A symmetric matrix with real elements can always be represented in the form S″ = ODO^-1, where O is an orthogonal matrix and

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References in periodicals archive

Let A [member of] [R.sup.nxn] be a nonsingular symmetric matrix, and let V [member of] [R.sup.nxs].

THE EXTENDED GLOBAL LANCZOS METHOD FOR MATRIX FUNCTION APPROXIMATION

(A1) There exist a symmetric matrix B and positive constants [delta] and [gamma] such that

STABILITY AND BOUNDEDNESS IN VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH DELAY

while the symmetric matrix [E.sub.ji] is gauge-invariant and is the symmetric square-root of the metric [g.sub.ij].

Dreibein as Prepotential for Three-Dimensional Yang-Mills Theory

It should be noted that [C.sup.T]C is a real symmetric matrix, the condition number is easy to obtain.

The Morbidity of Multivariable Grey Model MGM(1, m)

Let P [member of] [R.sup.(n-r)x(n-r)] be a nonzero symmetric matrix. For the symmetric matrix [mathematical expression not reproducible] [V.sup.T] we have [AX.sub.0] = [O.sub.m,n] and the proof of Lemma 3 is complete.

The Structure of Symmetric Solutions of the Matrix Equation AX = B over a Principal Ideal Domain

where [a.sub.i], i = 1, ..., p, [b.sub.j], j = 1, ..., q are arbitrary complex numbers, X is an m x m complex symmetric matrix, [C.sub.[kappa]](X) is the zonal polynomial of m x m complex symmetric matrix X corresponding to the ordered partition k = ([k.sub.1], ..., [k.sub.m]), [k.sub.1] [greater than or equal to] ...

Properties of matrix variate confluent hypergeometric function distribution

SDP problems arise from the well-known linear programming problems by replacing the vector of variables with a symmetric matrix and replacing the non-negativity constraints with positive semidefinite constraints.

Application of semidefinite programming to truss design optimization/Santvaros optimizavimo uzdaviniu sprendimas taikant pusiau apibrezta programavima

Theorem 2: A matrix [LAMBDA] is Hurwitz, that is, Re[[lambda].sub.i] [less than or equal to] 0 for all eigenvalues of [lambda], if and only if for any given positive-definite symmetric matrix Qthere a positive-definite symmetric matrix Pthat satisfies the Lyapunov equation

The error-squared controller: a proposed for computation of nonlinear gain through Lyapunov stability analysis/O controlador de erro- quadratico: uma proposta para o calculo do ganho nao linear atraves da analise da estabilidade de Lyapunov

matrix C reduces to a positive definite symmetric matrix P; as a result, (10) reduces to (25).

An LMI based criterion for global asymptotic stability of discrete-time state-delayed systems with saturation nonlinearities

The symmetric block component of a symmetric matrix is denoted by *.

A robust nonlinear observer for a class of neural mass models

The commonly used approach for spectral clustering link data is to obtain a symmetric matrix [bar.A] from the original adjacency A and then to apply spectral clustering techniques to [bar.A].

Digraph spectral clustering with applications in distributed sensor validation