Final answer:
To transform the quadratic form into one with no cross-product term, we diagonalize the matrix of coefficients by finding its eigenvalues and eigenvectors. The eigenvectors form the columns of the transformation matrix P. After applying x = Py, the quadratic form simplifies to just the sum of squares of the new variables.
Step-by-step explanation:
The problem asks us to make a change of variable, x = Py, that transforms the quadratic form 3x₁² + 10x₁x₂ + 3x₂² into a new quadratic form with no cross-product term. We're looking for a matrix P such that the quadratic form becomes a simple sum of squares.
The given quadratic form can be written in matrix notation as:
[x₁ x₂] * [[3 5] [5 3]] * [x₁ x₂]T
Where [[3 5] [5 3]] is the matrix of coefficients. To remove the cross-product term, we need to diagonalize this matrix. Diagonalization can be done by finding the eigenvalues and eigenvectors of the matrix. Once we have the eigenvectors, we can use them as columns to form the matrix P.
After finding matrix P, the transformation x = Py is applied, resulting in a new quadratic form that consists of only squares of the new variables with no cross-product term, typically of the form y₁² + dy₂², where d is the second eigenvalue.