Eigenvalue Problem and Spectral Decomposition of Second-Order Tensors

  • Mikhail ItskovEmail author
Part of the Mathematical Engineering book series (MATHENGIN)


So far we have considered solely real vectors and real vector spaces. For the purposes of this chapter an introduction of complex vectors is, however, necessary. Indeed, in the following we will see that the existence of a solution of an eigenvalue problem even for real second-order tensors can be guaranteed only within a complex vector space. In order to define the complex vector space let us consider ordered pairs \(\left\langle \varvec{x},\varvec{y}\right\rangle \) of real vectors \(\varvec{x}\) and \(\varvec{y} \in \mathbb {E}^n\). The sum of two such pairs is defined by [18]
$$ \left\langle \varvec{x}_1,\varvec{y}_1\right\rangle + \left\langle \varvec{x}_2,\varvec{y}_2\right\rangle =\left\langle \varvec{x}_1+\varvec{x}_2,\varvec{y}_1 + \varvec{y}_2\right\rangle .$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Continuum MechanicsRWTH Aachen UniversityAachenGermany

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