i was reading goldstein (oscillations)(adsbygoogle = window.adsbygoogle || []).push({});

in it, it is said that

Va= (lambda)*Ta

where V is the potential energy matrix, T is the kinetic eneregy matrix, lambda is an eigenvalue andais the corresponding eigenvector(of displacement from equilibrium)

and it is said that the matrix of eigenvectorsAdiagonalises the matrix V through a congruence transformationA'VA = L

where L is the diagonal matrix with its diagonal elements as eigenvalues. and A' is the transpose of A. this equation is solved by taking

|V-LI| = 0

here I is the identity matrix and 0 is the zero matrix.

but this secular equation is allowed only when the matrix A diagonalises V through a similarity transformation, isn't it?

it is the equation

Va= (lambda)*a

which will yield

|V-LI| = 0and

not the equationVa= (lambda)*Taisn't it?

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# Congruence transformation

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