In this paper, we derive transform formulae for linear functionals of affine processes and their bridges whose state space is the set of positive semidefinite d x d matrices. Particularly, we investigate the relationship between such transforms and certain integral equations. Our findings extend and unify the well known results of Cuchiero et al. (2011)  and Pitman and Yor (1982) , who analysed affine processes on positive semidefinite matrices and transforms of linear functionals of squared Bessel processes, respectively. We are, then, able to derive analytic expressions for Laplace transforms of some functionals of Wishart bridges. (C) 2013 Elsevier B.V. All rights reserved.