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Diagonalize symmetric matrix

  • Thread starter Locoism
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  • #1
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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 = PTAP 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
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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618

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 = PTAP 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.
 

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