Non-Hermitian (NH) systems exhibit intricate spectral topology and feature biorthogonal left and right eigenstates, resulting in distinct NH phenomena, such as the NH skin effect and ultraspectral sensitivity. However, conventional techniques fall short in directly measuring complex-valued spectra and biorthogonal eigenstates, particularly in higher-dimensional NH systems where geometry and boundary conditions become pivotal. Here, we present a Green's function-based method, which enables direct measurement of these fundamental quantities in arbitrary wave-based NH lattices. By capturing amplitude and phase responses for all pump-probe configurations experimentally, we acquire the full Green's function in two-dimensional non-Hermitian acoustic crystals. Utilizing this technique, we first demonstrate eigenstate skewness and spectral collapse in nonreciprocal lattices. Additionally, by controlling geometry and boundary conditions in reciprocal lattices, we observe spectral transitions—a unique hallmark of higher-dimensional NH systems. Our results not only render complex spectral topology and left eigenstates experimentally accessible and practically meaningful, but also establish a universal and versatile framework for exploring complex spectral features and NH dynamics across diverse physical systems.

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• Acoustic metamaterials
• Acoustics
• Phononic crystals
• Skin effect
• Topological phases of matter
• Non-Hermitian systems
• Topological insulators