Question

Quadratic forms arise often in geometry, physics and engineering and it is  desirable to reduce these to canonical (standard) forms. Reduce the quadratic form:

Q = 3x12 + 5 x22 + 3x32 - 2x2 x3 + 2x1 x3 - 2 x1 x2 to a standard form by making an approximate change in variables X = MY , where M is an  orthogonal matrix.  i) Write Q as X T AX .  ii) Find the Eigenvalues of A. Is the quadratic form positive definite?  iii) Find Eigenvectors corresponding to the Eigenvalues. Construct an orthogonal  modal matrix M to the matrix.  iv) X = MY is the required transformation. Write the relation between the old  and new system of coordinates. Write the canonical form for Q.  v) If Q = 1 what kind of quadric surface can it represent?