\(l^{p/2,q}\)-Singular Values of a Real Partially Symmetric Rectangular Tensor

It is known that the strong ellipticity condition of a general non-linearly elastic material can be equivalently transformed into the positive definiteness of a real (p, q)-th order \((m\times n)\) $(m\times n)$ -dimensional partially symmetric rectangular tensor \(\mathscr {A}\) $\mathcal{A}$ with p and q even. Furthermore, \(\mathscr {A}\) $\mathcal{A}$ is positive definite if and only if all of its \(l^{k,s}\) $l^{k,s}$ -singular values are positive, where \(k,s\geqslant 2\) $k,s⩾2$ are even. To determine the positive definiteness of \(\mathscr {A}\) $\mathcal{A}$ , we introduce its \(l^{p/2,q}\) $l^{p/2,q}$ -singular values by letting \(k=p/2\) $k=p/2$ and \(s=q\) $s=q$ . Subsequently, we adopt two approaches to this problem. The first approach is to construct an interval with parameters that contains all \(l^{p/2,q}\) $l^{p/2,q}$ -singular values. By optimizing these parameters, we obtain an optimal interval, which yields a criterion for the positive definiteness of \(\mathscr {A}\) $\mathcal{A}$ . The second approach is to develop two direct methods for computing all \(l^{p/2,q}\) $l^{p/2,q}$ -singular values/vectors of \(\mathscr {A}\) $\mathcal{A}$ for the specific cases where \(p=4\) $p=4$ , \(q=2\) $q=2$ , or 4, and \(m=n=2\) $m=n=2$ . These two methods are based on a case analysis of the \(l^{p/2,q}\) $l^{p/2,q}$ -singular vectors. Finally, we verify the correctness and effectiveness of the derived interval, criterion, and direct methods via a numerical example.