The open question in [J. Comput. Appl. Math. 221 (2008) 150–157] is “Is there a practical situation where a diffusion tensor \(D\in\mathbb{S}^{[2,3]}\) is not positive definite (PD)? If so, what can we do in such a case?”. This problem arises from research on computing all D-eigenpairs of a diffusion kurtosis tensor \(\mathcal{W}\in\mathbb{S}^{[4,3]}\) in medical diffusion kurtosis imaging. It can be phrased as follows: How can we compute all D-eigenpairs of \(\mathcal{W}\) when \(D\) is not a PD matrix? Here, \(\mathbb{S}^{[m,n]}\) is the set of all \(m\) th-order \(n\) -dimensional real symmetric tensors. In this paper, we address this problem for a tensor \(\mathcal{W}\in\mathbb{S}^{[m,n]}\) and a positive semi-definite (PSD) matrix \(D\in\mathbb{S}^{[2,n]}\) . When \(D\) is PD, we propose a Z-eigenpair-based method to compute all D-eigenpairs of the tensor pair \((\mathcal{W},D)\) . When \(D\) is PSD with zero eigenvalues, we first regularize it to a PD matrix \(D(t)\) (with parameter \(t > 0\) ), then compute all D-eigenpairs \((\lambda(t),\textbf{x}(t))\) of \((\mathcal{W},D(t))\) using this Z-eigenpair-based method, and finally refine them via Newton’s method to obtain all D-eigenpairs \((\lambda,\textbf{x})\) of \((\mathcal{W},D)\) .
