Show that the quadratic form 4x^2 + 2y^2 + 2z^2 − 2xy + 2yz − 2zx can be written as ~xT V ~x where ~x is a vector and where V is a symmetric matrix. Find the eigenvalues of V . Explain why, by rotating the
axes, the quadratic form may be reduced to the simple expression Ax'^2 + By'^2 + Cz'^2; what
are A, B, C?
I have done up to having found the eigenvalues, but am very confused about rotation matrices. Please help!
I presume, since you thought to find eigenvalues and eigenvectors, that you know that any symmetric matrix (and the matrix representing a quadratic is always symmetric) can be diagonalized: , where D is the diagonal matrix having the eigenvalues on the diagonal and P is a matrix having the eigenvalues as columns. If you choose the eigenvectors to be orthonormal (i.e. of unit length- they will necessarily be perpendicular), then P is the "rotation" matrix; it represents the rotation to take the xy-axes to the principal axes of the conic section defined by the quadratic.