Geometrical interpretation of the simultaneous diagonalization of two quadratic forms
P. K. Aravind · American Journal of Physics · 1989
The theory of small oscillations, when cast in matrix form, leads to an eigenvalue problem involving the simultaneous diagonalization of two real symmetric matrices. While a geometrical interpretation of this process exists, its visualization is generally hampered by the multidimensional nature of the coordinate transformations involved. I consider a model problem with just two degrees of freedom for which the coordinate transformations become planar and therefore easily visualizable. A set of plane diagrams is provided showing how the diagonalization is achieved for the model problem considered.