Can you please find the Eigenvalues and eigenvectors of this expression?

Question

p(x,y,z) = 3x² + y² +2z² - 2xy +2xz - 2yz

Answer 1

So you are expressing this quadratic form as X^TAX where X is vector (x,y,z) and A is a 3x3 symmetric matrix.

The x²,y², and z² coefficients are on the main diagonal and 1/2 of the value of the mixed terms will be in each spot corresponding to the variables. eg. -2xy implies that a₁₃ and a₃₁ will be -1:

Your matrix 3 -1 1

-1 1 -1

1 -1 2

Find the eigenvalues by finding the determinant of A-λI and equating to 0 (main diagonal is 3-λ,1-λ,2-λ)

You can use any row to expand or you can basketweave....

(3-λ)(2-λ)(1-λ)+1+1 -(2-λ)-(1-λ)-(3-λ) = 0

There is no joy here. I solved for eigenvalues numerically using this equation and they do correspond to the matrix that yields the quadratic form as written. (4.21432, 1.46081, .324869)

If you want the eigenvectors, you solve for the vector that, when multiplied by A-λ_iI, corresponds to each eigenvalue. From Wolfram alpha, I got vectors

corresponding to the eigenvectors listed. Good luck!