Question

if the Eigen values of the matrix A of a quadratic form are 2,-4 & 1 then write it's signature.​

Answer 1

Answer:

Here's a start. You want a symmetric matrix. The diagonal elements correspond to the squares, and the non-diagonals to half the mixed monomials:

2x2+6y2+2z2+8xz=⎛⎝⎜xyz⎞⎠⎟T⎛⎝⎜204060402⎞⎠⎟⎛⎝⎜xyz⎞⎠⎟

With A the square matrix above, the characteristic polynomial is then

p(λ)=det(A−λI)=⎛⎝⎜λ−20−40λ−60−40λ−2⎞⎠⎟=(λ−2)2(λ−6)+16(λ−6)=(λ+2)(λ−6)2

so there should be one eigenvector for λ=−2 and two for λ=6.

$thanks$