Two (real or complex) \(m \times n\) matrices A and B are called parallel (respectively, triangle equality attaining, abbreviated TEA) if \(\begin{aligned} \Vert A + \mu B\Vert = \Vert A\Vert + \Vert B\Vert \end{aligned}\) holds for some scalar \(\mu \) with \(|\mu | = 1\) (respectively, \(\mu = 1\) ), where \(\Vert \cdot \Vert \) denotes the spectral norm. We investigate linear maps T acting on \(m \times n\) matrices that preserve parallel (resp. TEA) pairs, meaning T(A) and T(B) remain parallel (resp. TEA) whenever A and B are parallel (resp. TEA). When \(m,n \ge 2\) and \((m,n) \ne (2,2)\) , we show that a nonzero linear map T preserves TEA pairs if and only if T is a positive scalar multiple of a linear isometry. Specifically, T must take one of the following forms: (1) \(A \mapsto \gamma UAV\) , or
(2) \(A \mapsto \gamma UA^{{\textrm{t}}} V\) (with \(m = n\) in this case),
where \(\gamma > 0\) is a scalar and U, V are unitary (or real orthogonal) matrices of appropriate sizes. For parallel pairs, the preserving maps include the above forms along with (3) \(A \mapsto f(A) Z\) ,
where f is a linear functional and Z is a fixed matrix. The \(2 \times 2\) case exhibits richer structure: there exist linear maps preserving parallel or TEA pairs that do not conform to (1), (2), or (3). A complete classification of such maps is established using intricate matrix-theoretic computations and techniques from matrix group theory.
