Geometric algebra was initiated by W.K. Clifford over 130 years ago. It unifies all branches of physics, and has found rich applications in robotics, signal processing, ray tracing, virtual reality, computer vision, vector field processing, tracking, geographic information systems and neural computing. This tutorial explains the basics of geometric algebra, with concrete examples of the plane, of 3D space, of spacetime, and the popular conformal model. Geometric algebras are ideal to represent geometric transformations in the general framework of Clifford groups (also called versor or Lipschitz groups). Geometric (algebra based) calculus allows, e.g., to optimize learning algorithms of Clifford neurons, etc.

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neural networks. Once energy functions are constructed forRecently models of neural networks that can deal with complex numbers, complex-valued neural networks, have been proposed and several studies on their abilities of information processing have been done. In this paper we investigate existence conditions of energy functions for a class of fully connected complex-valued neural networks and propose an energy function, analogous to those of real-valued Hopfield-type neural networks. It is also shown that, similar to the real-valued ones, the energy function enables us to analyze qualitative behaviors of the complex-valued neural networks. We present dynamic properties of the complexvalued neural networks obtained by qualitative analysis using the energy function. A synthesis method of complexvalued associative memories by utilizing the analysis results is also discussed.