Let A [member of] [R.sup.nxn] be a nonsingular

symmetric matrix, and let V [member of] [R.sup.nxs].

(A1) There exist a

symmetric matrix B and positive constants [delta] and [gamma] such that

while the

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

It should be noted that [C.sup.T]C is a real

symmetric matrix, the condition number is easy to obtain.

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.

where [C.sub.[kappa]](X) is the zonal polynomial of mxm complex

symmetric matrix X corresponding to the ordered partition [kappa] = ([k.sub.1], ..., [k.sub.m]), [k.sub.1] [greater than or equal to] ...

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] ...

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.

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

matrix C reduces to a positive definite

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

The symmetric block component of a

symmetric matrix is denoted by *.

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].