Find the new quadratic form?

Question

Classify the quadratic form below. Then make a change of​ variable, x=Py​, that transforms the quadratic form into one with no​ cross-product terms. Write the new quadratic form.

-X²₁-3X²₂-7X²₃-7X²₄+2X₁X₂+2X₁X₃+2X₁X₄+8X₃X₄

What is the quadratic form ?

The new quadratic form is

y^TDy= _________

Answer 1

Since you need to find the transformed quadratic form it suffices to find the eigenvalues of the quadratic form in x coordinates:

Q = (1,-1,-1,-1),(-1,3,0,0),(-1,0,7,-4),(-1,0,-4,7))

The characteristic polynomial can be found by evaluating the determinant |Q - λI| and its roots are the eigenvalues. In this case the eigenvalues are:

E = (11,4,3,0)

While the eigenvectors are:

V₁₁ = (0,0,-1,1)

V₄ = (-1,1,1,1)

V₃ = (0,-2,1,1)

V₀ = (3,1,1,1)

The matirx P is then P = (V_11,V₄,V₃,V₀) and in this variables the quadratic form is:

Q_y = ((11,0,0,0),(0,4,0,0),(0,0,3,0),(0,0,0,0))

y^t Q y = 11 y₁² + 4 y₂² + 3 y₃²