Question

Bring the quadratic form given by coordinate representation below in some basis in a real vector space V,dimV=n to a canonical form. Find its rank and positive and negative indices. Write the corresponding coordinate transformation:

(a) q(x)=x²₁+4x₁x₂+2x₁x₃+2x²₂+8x₂x₃,(n=3) [5]
(b) q(x)=x₁x₂+2x₂x₃−3x₃x₄,(n=4). [10]

Answer 1

To bring the quadratic form to canonical form, find the eigenvalues and eigenvectors of the quadratic form matrix.

Step-by-step explanation:

To bring the quadratic form given by the coordinate representation to canonical form, we need to find the orthogonal basis in which the quadratic form becomes diagonal. We can achieve this by finding the eigenvalues and eigenvectors of the given quadratic form matrix.

For part (a), the quadratic form is q(x) = x²₁+4x₁x₂+2x₁x₃+2x²₂+8x₂x₃. The matrix representation of this quadratic form is Q = [[1, 2, 1], [2, 0, 4], [1, 4, 0]]. We can find its eigenvalues and eigenvectors to form the coordinate transformation matrix.

For part (b), the quadratic form is q(x) = x₁x₂+2x₂x₃−3x₃x₄. The matrix representation of this quadratic form is Q = [[0, 1, 0, 0], [1, 0, 3, 0], [0, 3, 0, -2], [0, 0, -2, 0]]. We can find its eigenvalues and eigenvectors to form the coordinate transformation matrix.