  This study develops identification methods for linear continuous-time
symmetric systems, such as electrical network systems, multi-agent network
systems, and temperature dynamics in buildings. To this end, we formulate three
system identification problems for the corresponding discrete-time systems. The
first is a least-squares problem in which we wish to minimize the sum of
squared errors between the true and model outputs on the product manifold of
the manifold of symmetric positive-definite matrices and two Euclidean spaces.
In the second problem, to reduce the search dimensions, the product manifold is
replaced with the quotient set under a specified group action by the orthogonal
group. In the third problem, the manifold of symmetric positive-definite
matrices in the first problem is replaced by the manifold of matrices with only
positive diagonal elements. In particular, we examine the quotient geometry in
the second problem. We propose Riemannian conjugate gradient methods for the
three problems, and select initial points using a popular subspace method. The
effectiveness of our proposed methods is demonstrated through numerical
simulations and comparisons with the Gauss--Newton method, which is one of the
most popular approach for solving least-squares problems.
