]> 32.9 Application to Quadratic Forms and Spring Systems

32.9 Application to Quadratic Forms and Spring Systems

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. (Thus we can change $x 2$ to $4 u 2$ by setting $u = x 2$ .)

This allows us to diagonalize two distinct quadratic forms simultaneously. You can make the matrix of one into an identity matrix, and then diagonalize the other.

This is in contrast to what happens with transformations. Two transformations must commute to be simultaneously diagonal with the same basis (obviously necessary since all diagonal matrices commute with one another).

(This statement is proven as follows:

Diagonalize the matrix $M$ . You can then observe that the condition that $M$ and $N$ commute 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 above 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 (each of which can represent a complex combination of original coordinates). The corresponding eigenvalues determine the "normal modes" of the system.