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Another place in which matrices have appeared in previous chapters was in the
discussion of the behavior of functions of several variables at a critical point
(at which the gradient of the function is the 0 vector.) We then noticed that the behavior of the function could be described by the
matrix of second derivatives of the function at that point. This is the matrix
whose ij-th element is the second partial derivative of the function with respect
to the i-th and j-th variable. Symmetric matrices each have a an orthonormal basis of real eigenvectors as
we proved above obtainable by an orthogonal transformation. If we examine the structure of the matrix for the form using this basis, we
find that it is diagonal, and so the conditions for an extremum become simple:
if all the eigenvalues of the second derivative matrix have
the same sign the function has a local maximum or minimum, with a minimum when
they are all positive. There is a saddle if the signs are mixed, and you
must sometimes look at higher derivatives when some of the eigenvalues are 0's. When talking about the matrix of second derivatives we are really talking about
the quadratic form which describes the quadratic terms in the Taylor series expansion
of our function about the critical point. If we focus on quadratic forms we realize that we can use a wider class of
transformations in order to change their appearance than we can when dealing with
transformations. Thus we can make changes of scale of the individual variables
to make any positive diagonal quadratic into one that has all (non-zero) eigenvalues
the same. (This statement is proven as follows. Diagonalize the matrix M. You can then
observe that the condition that M and N ccmmute is the condition that all the
off diagonal elements of N say the ij-th link indices whose diagonal elements
of M are the same. Thus if the ij-th entry of N is non-zero then the i-th and
j-th eigenvalues of M must be the same if N and M are to commute. If they are
the same then as far as diagonalizing N is concerned, the problem breaks up into
parts for each eigenvalue of M; and for each part M is a multiple of the identity
matrix and will stay diagonal when the corresponding block of N is diagonalized.) Given a system of springs and masses, there will be one quadratic form that represents the kinetic energy of the system in terms of momentum variables, and another which represents the potential energy of the system in position variables. The remarks about tell us that it is always possible to choose a normalization and basis of coordinates so that both of these forms are diagonal. This means that the entire system can be analyzed as a bunch of independent simple one dimensional springs. The corresponding eigenvalues determine the "normal modes" of the system. |