By using the linear structure theory of Magnus (12), this work proposes an alter- native way to James (11) for obtaining the Laplace-Beltrami operator, who has the zonal polynomials of positive definite matrix argument as eigenfunctions, in partic- ular, an explicit expression for the matrix G(v(X)), which appears in the metric differential form (ds)2=dv0(X)G(v(X))dv (X), is obtained; also, the invariance f (ds)2 under congruence transformations is proved. Explicit forms for (ds)2 and G(v(X)) are also shown under the spectral decomposition X=HY H0. In a newapproach -apart from the classical theory of James (11)- a differential metric de- pending on the Moore-Penrose inverse is proposed for the space of m£m positive semidefinite matrices. As in the definite case, the Laplace-Beltrami operator for the calculation of zonal polynomials of positive semidefinite matrix argument is de- rived. In a parallel way the invariance of (ds)2 is shown and explicit expressions for the metric and the matrixG (¢) are obtained in terms of X and its spectral decomposition. Finally, an efficient computational method for calculating the zonal polynomials of positive semidefinite matrices are preseted.