  A tensor in applied mathematics is usually defined as a multidimensional
array of numbers. This presumes a choice of basis in $\mathbb{R}^n$ or in some
other vector space, and tensorial concepts are defined accordingly. In this
article we define eigenvalues, eigenvectors, singular values, and singular
vectors of a tensor intrinsically, without reference to a basis. The basis-free
approach has several advantages. First, it shows more clearly the relationship
between tensor analysis and areas of pure mathematics such as abstract algebra,
differential topology, and algebraic geometry. Second, it obviates the need to
prove that a concept defined in terms of coordinates is independent of the
choice of basis. Third, an intrinsic definition is usually conceptually
simpler. As illustrations we show how Morse theory from differential topology
can be used to analyze eigenvalues and eigenvectors of a symmetric tensor. We
also reprove a few results that are obvious in the basis-free approach, but not
otherwise.
