Dynamics of two by two symmetric matrices of trace zero

Abstract

In this paper, we describe the entire structure of the vector space $Sym_2^0$ of all symmetric matrices of size $2$ having trace zero. This is motivated by the geometrical interpretation of any arbitrary element of $Sym_2^0$. We further study the orbits and stable sets of these elements. As an application of the obtained structure of $Sym_2^0$, we obtain the symmetric matrices of size $2$, trace of whose product with any trace zero symmetric matrix is zero. Finally some well known trigonometric formulas are interpreted geometrically incorporating the anatomy of $Sym_2^0$.

Figures (2)

• Figure 1: Image of a point $P_0$ under the map $\lambda R_{\theta}$ for some values of $\lambda$
• Figure 2: Convergence of the sequence $\{(\lambda R)^n(P_0)\}_{n\geq 0}$ for $0<\mid \lambda \mid <1$

Theorems & Definitions (23)

• Remark 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• proof
• Proposition 2.5
• proof
• Theorem 2.6