9.58(b), §2 Solution

q = 2 x² - 6 xy + 10 y²

Find the matrix of coefficients:

Eigenvalues:

There are two roots: and

The eigenvector corresponding to satisfies

Taking v_1y = 1, then v_1x = 3, giving an eigenvector (3,1). Normalizing this vector to length one gives:

The eigenvector corresponding to satisfies

Taking v_2y = 3, then v_2x =-1, giving after normalization:

Since , the new axes are rotated counter clockwise from the old:

In the new coordinates,

Note that lines of constant q are now seen to be elliptic.

Important note: It is seen that the quadratic form is always positive for nonzero . Symmetric matrices for which this is true are called positive definite. They have all positive eigenvalues. Similarly, if all eigenvalues are negative, a symmetric matrix is called negative definite. If all eigenvalues are positive or zero, it is called positive semi-definite.

Finite element codes for structures typically produce positive definite matrices, as do many other physical applications, such as the kinetic energy of a solid body. Definite matrices are typically easier to deal with in numerical applications than general matrices. For example, no pivoting is needed in the Gaussian elimination involving a definite matrix.