Symmetric Matrix


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symmetric matrix

[sə′me·trik ′mā·triks]
(mathematics)
A matrix which equals its transpose.

Symmetric Matrix

 

a square matrix S = ||sik|| in which any two elements that are symmetrically located with respect to the principal diagonal are equal: sik = ski, 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,…, λn 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

References in periodicals archive ?
If X is an m x m symmetric matrix, and R is an m x m symmetric positive definite matrix, then the eigenvalues of RX are same as those of [R.
We can take skew- symmetric matrix A and 3 x 1 column matrix C as below
where the skew symmetric matrix Q is defined by Q = ([Q.
It is well known that since A is a real symmetric matrix all its eigen values must be real.
These condition are described in terms of certain symmetric matrix over [R.
Where CA is an asymmetric matrix, CS is a symmetric matrix, and CTA is the transposed asymmetric matrix.
T] = PA, or symmetric P-skew symmetric matrix if A = [A.
Where the elements of the (5x5) symmetric matrix [summation](M) are given by Cov([m.
Of course, the functional G would not exist if A was not a symmetric matrix.

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