  We study orthogonal decompositions of symmetric and ordinary tensors using
methods from linear algebra. For the field of real numbers we show that the
sets of decomposable tensors can be defined be equations of degree 2. This
gives a new proof of some of the results of Robeva and Boralevi et al.
Orthogonal decompositions over the field of complex numbers had not been
studied previously; we give an explicit description of the set of decomposable
tensors using polynomial equalities and inequalities, and we begin a study of
their closures. The main open problem that arises from this work is to obtain a
complete description of the closures. This question is akin to that of
characterizing border rank of tensors in algebraic complexity. We give partial
results using in particular a connection with approximate simultaneous
diagonalization (the so-called "ASD property").
