SYMMETRIC MATRICES AND QUADRATIC FORMS

Orthogonally Diagonalizable Symmetric Matrices and Eigenvalues How to Orthogonally Diagonalize Quadratic Forms Definiteness Principal Axes Ellipses and Hyperbolas

Ellipses and Hyperbolas How to Examples Exercise How to: Consider the quadratic form in R2 defined by the equation:      q(x1, x2) = a*x1^2 + b*x1*x2 + c*x2^2 = 1 Define A to be the symmetric matrix A = [a b/2; b/2 c]. Let lambda1 and lambda2 be the eigenvalues associated with matrix A. Based on these eigenvalues, we can say a lot about what the quadratic form will look like geometrically. Case 1: lambda1 > 0, lambda2 > 0 Here we get an ellipse with principal axes deteremined by the eigenvectors of lambda1 and lambda2. Case 2: lambda1 > 0, lambda2 < 0 (or vice-versa) Here we get a hyperbola with principal axes determined by the eigenvectors of lambda1 and lambda2.      Examples: Exercise: