Showing papers by "Ingram Olkin published in 1965"

Monotonicity of ratios of means and other applications of majorization

Abstract: : Monotonicity of a ratio can be viewed as a form of total positivity. The theory of total positivity is exploited to obtain more general results. The proof of monotonicity is based on a theorem giving sufficient conditions for majorization. Several other applications of the majorization theorem are given. One application concerns a stochastic comparison between a function of order statistics from a distribution with increasing failure rate average and the same function of the order statistics from the exponential distribution. Another application is to a comparison between the condition number of a positive definite matrix and the condition number of a polynomial in the matrix.

Norms and inequalities for condition numbers

Abstract: : The condition number c sub phi of a nonsingular matrix A is defined by c sub phi (A) = phi (A) phi (A superscript -1) where ordinarily phi is a norm. It was shown by J. D. Riley that if A is positive definite, c sub phi (A + kI) = or 0 and phi squared (A) is the maximum eigenvalue of AA* or phi squared (A) = Tr AA*. In this paper it is shown more generally that c sub phi (A + B) = or < c sub phi (B) when phi satisfies phi (U) = or < phi (V) if V-U is positive definite and when A,B are positive definite satisfying c sub phi (A) = or < c sub phi (B). Some related inequalities are also obtained. As suggested by Riley, these results may be of practical use in solving a system Ax = d of linear equations when A is positive definite but ill-conditioned. (Author)

On the bias of functions of characteristic roots of a random matrix

Abstract: Let Z = (zij) be arandomp xp symmetricmatrixwith EZ = A, i.e. Ezij = aj (i,j= 1,... ,p). An application to the theory of response surface estimation led van der Vaart (1961) to consider the expectation-bias and median-bias of the characteristic roots of Z as estimators of the characteristic roots of A. Denote the (real) characteristic roots of a matrix X by A(X) with the ordering A1(X) > ... > Ap(X). Van der Vaart proved that if Z is a symmetric matrix, then EA1(Z) > A1(A) and EAp(Z) O} = P{trC(Z -A) A1(A)} > 1 and P{Ap(Z) 2. If absolute continuity is assumed, these bias inequalities become strict. The purpose of this paper is to show how to generate a wide class of inequalities between EA(Z) and A(EZ). In particular, some of the results of van der Vaart (1961) can be strengthened by a weakening of the assumption that Z be a symmetric matrix. When Z is symmetric, the expectation-bias and median-bias may also be obtained for partial sums of the roots, and when Z is positive definite, the expectation-bias is obtained for more general functions of the roots. Expectation-bias for the roots of the determinantal equation IZl Z21 = 0 is presented; of particular interest is the determinantal equation for the canonical correlations. Inequalities for the median-bias of certain linear combinations of the roots are also developed. These inequalities are obtained as direct consequences of known results concerning the convexity of scalar functions of a matrix. An extension of Jensen's inequality to convex matrix functions using the Loewner ordering for matrices is also given; this extension then serves as a source of other inequalities.