If the n X n Hermitian matrix
A has a principal submatrix Aj satisfying the following:
the determinant of a Hermitian matrix
is defined by putting det A = [rdet.sub.i] A = [cdet.sub.i] A for all i = 1, ..., n.
Actually, (37) implies that g([[phi].sub.k]) is the eigenvector associated with the minimum eigenvalue of the Hermitian matrix
For a Gaussian random Hermitian matrix
[rho](M) [varies] exp - (1/2[[sigma].sup.2])Tr([M.sup.2]), the eigenvalue density obeys Wigner's semicircle law:
For a Hermitian matrix
, the Moore's determinant is defined by specifying a certain ordering of the factors in the n!
A Hermitian matrix
polynomial Q([lambda]) is definite if and only if any two (and hence all) of the following properties hold:
Banking on the fact that K[I.sup.[phi].sub.S(p,v)] is a positive Hermitian matrix
, while referring to (21), we are sure that all eigenvalues of [lambda] are greater than or equal to [SNR.sup.-1] and therefore are strictly positive.
Note that, for positive semidefinite Hermitian matrix
[S.sup.H]S, its condition number doubles cond(S); that is,
Since (PA(e) + [(A[(e)).sup.H] P).sup.H] = PA(e) + [(A(e)).sup.H] P, then PA(e) + [(A(e)).sup.H] P is a Hermitian matrix
. At the same time, since PA(e) + [(A(e)).sup.H] P = -Q, P is a real symmetric positive definite matrix, and Q is a positive definite matrix, then [[beta].sup.H] P[beta] > 0 and [[beta].sup.H] (-Q) [beta] < 0, so that
Finally, the notations d * and [lambda] * represent the main diagonal elements and eigenvalues of a Hermitian matrix
Gupta and Luz Estela Sanchez, "A class of integral identities with Hermitian matrix
We also denote by G the n x n hermitian matrix
[([g.sub.ij]).sup.n.sub.i,j=1], and set [1.sub.n] for the n x n identity matrix.