Given a positive and unitarily invariant Lagrangian L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document} defined in the algebra of matrices, and a fixed time interval [0,t0]⊂R\documentclass[12pt]{minimal}
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\begin{document}$${[0,t_0]\subset\mathbb R}$$\end{document}, we study the action defined in the Lie group of n×n\documentclass[12pt]{minimal}
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\begin{document}$${n\times n}$$\end{document} unitary matrices U(n)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{U}(n)}$$\end{document} by
S(α)=∫0t0L(α˙(t))dt,\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal{S}(\alpha)=\int_0^{t_0} \mathcal{L}(\dot\alpha(t))\,dt, $$\end{document}where α:[0,t0]→U(n)\documentclass[12pt]{minimal}
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\begin{document}$${\alpha:[0,t_0]\to\mathcal{U}(n)}$$\end{document} is a rectifiable curve. We prove that the one-parameter subgroups of U(n)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{U}(n)}$$\end{document} are the optimal paths, provided the spectrum of the exponent is bounded by π. Moreover, if L\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}$$\end{document} is strictly convex, we prove that one-parameter subgroups are the unique optimal curves joining given endpoints. Finally, we also study the connection of these results with unitarily invariant metrics in U(n)\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{U}(n)}$$\end{document} as well as angular metrics in the Grassmann manifold.
