Thread: Orthogonal Change of Variables

Find an orthogonal change of variables X=PY such that:

–2x1^2 + 2x1x2 - 2x2^2

takes the form:

Ay1^2 + By2^2

Sorry if this may be confusing. The numbers next to the variables should be subscripts. For instance x1^2 should be x subscript 1 to the power of 2.

Write the quadratic form as . Then find the eigenvalues and eigenvectors of the matrix and diagonalise it.

my problem is find an orthogonal change in variables x=py such that
4x1^2 + 10x1x2 + 4x2^2
takes the form ?x1^2 + ?x2^2
p=?
the question marks are blanks for answers, there should be subscripts and ^2 is squared

so i think i understand how to find p... you diagonalize and find the eigenvectors, but i dont know how to answer the x1^2 + x2^2 part.

Originally Posted by luccasaurus

my problem is find an orthogonal change in variables x=py such that
4x1^2 + 10x1x2 + 4x2^2
takes the form ?x1^2 + ?x2^2
p=?
the question marks are blanks for answers, there should be subscripts and ^2 is squared

so i think i understand how to find p... you diagonalize and find the eigenvectors, but i dont know how to answer the x1^2 + x2^2 part.

Step 1: Write the quadratic form as , where and .

Step 2: Diagonalise A, to get , where D is the diagonal matrix (-1 and 9 being the eigenvalues of A), and P is the orthogonal matrix whose columns are the corresponding normalised eigenvectors.

Step 3: Then the quadratic form is , where .