Tensors, as geometric objects that describe linear or multi-linear relations between geometric vectors, scalars and other tensors, have provided a concise mathematical framework for formulating and solving practical physics problems in various areas such as relativity theory, fluid dynamics, solid mechanics and electromagnetism, etc. The concept of tensors can be traced back to the works by Carl Friedrich Gauss (1777-1855), Bernhard Riemann (1826-1866) and Elwin Bruno Christoffel (1829-1900), etc., in the 19th century on differential geometry. It was further developed and analyzed by Gregorio Ricci-Curbastro (1853-1925), Tullio Levi-Civita (1873-1941), and others, in the very beginning of the 20th century. A mathematical discipline on tensor analysis gradually emerged and was even applied in general relativity by the great scientist Albert Einstein (1879-1955) in 1916.
While tensors such as piezoelectric tensors and elasticity tensors have been used in physics and mechanics for more than one century, the study on spectral properties of these tensors is still very new. The fundamental principle of Galileo Galilei (1564-1642) who has played a pioneer role in the scientific revolution of the seventeenth century and is regarded as the father of science, is to study the rules and insights of the nature, while mathematics is the basic tool in this process. Inspired by this principle, the mathematical analysis on spectral properties have been studied for tensors in liquid crystal study, piezoelectric effects, solid mechanics, quantum entanglement problems, etc. More spectral properties of tensors in physics and mechanics awaits being exploited.