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Orthogonal matrices are powerful mathematical tools that preserve vector lengths and angles under transformation—properties that mirror the symmetry and coherence found in coherent sound waves. In acoustic physics, this geometric fidelity ensures predictable echo patterns, enabling precise modeling of wave propagation. Like a perfectly aligned mirror reflecting light without distortion, orthogonal operators maintain structural integrity in wave systems, forming the backbone of stable acoustic simulations.

An orthogonal matrix $ Q $ satisfies $ Q^T Q = I $, meaning its rows and columns are orthonormal vectors. This preservation of inner products ensures that the dot product—and thus angles and magnitudes—remain unchanged after transformation. In sound wave propagation,

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