Let Z = (zij) be a random p × p symmetric matrix with EZ = A,i.e.Ezij = aij(i,j = l,...,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 λ(X) with the ordering λ1(X) ≥ ... ≥ λp(X). Van der Vaart proved that if Z is a symmetric matrix, then Eλ1(Z) ≥ λ1(A) and Eλp(Z) ≤ λp(A). With the additional assumption that the distribution of Z is symmetric about A, i.e. P(trC(Z-A) ≤ 0} = P{tvC(Z-A) ≤ 0}, for all matrices C = (cij) ≢ 0, he obtained similar inequalities for the median-bias, namely, P{λ1(Z) ≥ λ1(A)} ≥ 1/2 and P{λp(Z) ≤ λp(A)} ≥ 1/2. 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 Eλ(Z) and λ(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 ∣Z1 – θZ2∣ = 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.