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In this paper, we aim to introduce and characterize the numerical radius orthogonality of operators on a complex Hilbert space $${\mathcal {H}}$$ H which are bounded with respect to the seminorm induced by a positive operator A on $${\mathcal {H}}$$ H . Moreover, a characterization of the A -numerical radius parallelism for A -rank one operators is obtained. As applications of the results obtained, we derive some $${\mathbb {A}}$$ A -numerical radius inequalities of operator matrices, where $${\mathbb {A}}$$ A is the operator diagonal matrix whose each diagonal entry is a positive operator A on a complex Hilbert space $${\mathcal {H}}.$$ H .

A-Numerical Radius Orthogonality and Parallelism of Semi-Hilbertian Space Operators and Their Applications