We address the problem of detecting the number of complex exponentials and estimating their parameters from a noisy signal using the Matrix Pencil (MP) method. We introduce the MP modes and present their informative spectral structure. We show theoretically that these modes can be divided into signal and noise modes, where the signal modes exhibit a perturbed Vandermonde structure. Leveraging this structure, we propose a new MP algorithm, termed the SAMP algorithm, which has two novel components: (i) a robust, theoretically grounded model-order detection method, and (ii) an efficient estimation of the signal amplitudes. We show empirically that SAMP significantly outperforms the standard MP method in challenging conditions, with closely-spaced frequencies and low Signal-to-Noise Ratio (SNR) values, approaching the Cramer-Rao lower bound (CRB) for a broad SNR range. Additionally, compared with prevalent information-based criteria, we show that SAMP achieves comparable or better results with lower computational time, and lower sensitivity to noise model mismatch.