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This Mathcad document uses symmetry to simplify the evaluation of an eigenvalue problem by reducing a matrix to sparse form, a form with many more zeroes in the off diagonal positions. The sparse matrix is then transformed to block diagonal form with each sub matrix diagonalized separately. The technique can be used for any molecular system with symmetry. Group theory is used to set up the calculations without going through the proofs of the methods. Calculations like matrix multiplication, Taylor expansions, and derivatives are performed by symbolic evaluation in Mathcad. Experiencing the great reduction in the calculation helps students see the value of studying this important field of science. The level of understanding is typically graduate and maybe some undergraduate. The ability to use symmetry principles to get the vibrational frequencies and other information about molecules with symmetry is an important asset to any chemist. 
Equilibrium distance vectors, b1, b2, and b3 are displaced by the vibration
variables Δr1, Δr2, and Δr3 to produce vectors r1, r2, and r3 during a vibration.
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