Birkhoff-James orthogonality of linear operators on finite dimensional Banach spaces

In this paper we characterize Birkhoff-James orthogonality of linear operators defined on a finite dimensional real Banach space $\mathbb{X}.$ We also explore the symmetry of Birkhoff-James orthogonality of linear operators defined on $\mathbb{X}.$ Using some of the related results proved in this paper, we finally prove that $T \in \mathbb{L}(l_{p}^2) (p \geq 2, p \neq \infty)$ is left symmetric with respect to Birkhoff-James orthogonality if and only if $T$ is the zero operator. We conjecture that the result holds for any finite dimensional strictly convex and smooth real Banach space $\mathbb{X},$ in particular for the Banach spaces $l_{p}^{n} (p > 1, p \neq \infty).$