Lee [3], in 2018, gave necessary and sufficient conditions for a diagonal quadratic form \(\sum _{i=1}^{m} a_{i}X_{i}^{2}\) to represent every matrix in \(M_2(\mathbb {Z}),\) where \(a_i \in \mathbb {Z}\) \((1 \le i \le m)\) . The discriminant criterion given by Katre and Khule [2], in 1999, over the rings of integers \(\mathcal {O}\) of an algebraic number field K says that every \(2 \times 2\) matrix over the ring of integers \(\mathcal {O}\) of an algebraic number field can be written as a sum of squares over \(M_2(\mathcal {O})\) if and only if the discriminant of \(\mathcal {O}\) is odd. Let \(\mathbb {Z}[\omega ]\) be the ring of integers of the imaginary quadratic number field \(\mathbb {Q}(\sqrt{-3}),\) where \(\omega =\frac{-1+\sqrt{-3}}{2}\) . Let \(\mathcal {O}=\mathbb {Z}[\omega ],\) where \(\omega ^2+\omega +1=0.\) This ring is well-known in literature as the Eisenstein ring. Note here that the discriminant of the Eisenstein ring is \(-3,\) which is odd. Hence, it is expected that we might obtain a result like that of Lee [3] for the general quadratic form \(\sum _{i=1}^{m} a_{i}X_{i}^{2}\) (since \(a_i=1\) for all i gives the special case of matrices in \(M_2(\mathcal {O})\) being sum of squares). The aim is to obtain necessary and sufficient conditions to represent every matrix in \(M_2(\mathcal {O})\) by the diagonal quadratic form \(\sum _{i=1}^{m} a_{i}X_{i}^{2}\) i.e., obtain conditions for representing every \(2 \times 2\) matrix over the Eisenstein ring as a diagonal quadratic form.
