I would like some explanation on why we use the followingmethod: For a quadratic

Question

I would like some explanation on why we use the followingmethod:
For a quadratic form Q(x) = xT A x, where a change ofvariable is required,
1. Translate Q(x) to xT A x
2. Orthogonally diagonalise A
3. Change the variables x = Py
4. Go back to the start, write down all the equations (comes downto Q(x)=yTDy
5. Solve,Q(x)=ay12+by22
Now,since 'a' is just x11 of D and 'b' is just x22 of D, I wonder why Igo through all that trouble of orthogonal diagonalisation? And thenchange the variables around? Can't I just find the 's andPRESTO, that's y1 and y2 ?

Explanation / Answer

if a change of varibles is needed for the function in the formof f(x,y) or for a PDE f(uxx, uyy) andfor hairyer functions (mixed term ie xy oruxy) or functions that are/need to be solved by thefourier method (a method to orthongonalize to get nice less hairysols). so for just a plain ole change of varible problem youcan just "super-impose" a solution to get your eigenvalues andsolve for them. I hope I understand your question (im not that farin linear...) I hope my responce is kinda clear, as you can see I've takenPDE (partial differential equations) and the fourier method is aorthogonal representation of a matrix if you will (using that termvery losely).

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