Is there a way to diagonalize a symmetric matrix without using a calculator?

[Locoism]

Homework Statement

I need to diagonalize the matrix A=
1 2 3
2 5 7
3 7 11

The Attempt at a Solution

Subtracting λI and taking the determinant, the characteristic polynomial is
λ³ - 17λ² + 9λ - 1 (I have checked this over and over)

The problem now is it has some ugly roots, none that I would never be able to find without a calculator (which is sort of the objective here). Anyways, is there some other way to find P such that
P^-1AP = P^TAP is a diagonal matrix?

*edit* I would just add that the matrix A is the matrix associated to the quadratic form q(v) = x² + 5y² + 11z² + 4xy + 6xz + 14yz

 

[Dick]

Homework Statement

I need to diagonalize the matrix A=
1 2 3
2 5 7
3 7 11

The Attempt at a Solution

Subtracting λI and taking the determinant, the characteristic polynomial is
λ³ - 17λ² + 9λ - 1 (I have checked this over and over)

The problem now is it has some ugly roots, none that I would never be able to find without a calculator (which is sort of the objective here). Anyways, is there some other way to find P such that
P^-1AP = P^TAP is a diagonal matrix?

*edit* I would just add that the matrix A is the matrix associated to the quadratic form q(v) = x² + 5y² + 11z² + 4xy + 6xz + 14yz

It is true the roots are ugly. And there's no way to diagonalize that without some calculator assistance. On the other hand is that really the question? Do you want to show it's positive definite? Then you just need to show the eigenvalue equation has three positive roots. You can do that with calculus. Find the critical points of the eigenvalue equation. Basically, sketch a graph of it.