Rank Two Approximations of \(2 \times 2 \times 2\) Tensors over \(\mathbb{R}\)

We provide a coordinate-free proof that real \(2 \times 2 \times 2\) rank three tensors do not have optimal rank two approximations with respect to the Frobenius norm. This result was first proved in by considering the \({\text{GL}(V^1) \times \text{GL}(V^2) \times \text{GL}(V^3)}\) orbit classes of \({V^1 \otimes V^2 \otimes V^3}\) and the \(2 \times 2 \times 2\) hyperdeterminant. Our coordinate-free proof expands on this known result by developing a proof method that can be generalized more readily to higher dimensional \(n_1 \times n_2 \times n_3\) tensor spaces.