Let K be a positive definite 3 ×3 matrix. (a) Prove that the quadratic equation 𝐱^T K 𝐱=1 define

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Let K be a positive definite 3 ×3 matrix. (a) Prove that the...

Let $K$ be a positive definite $3 \times 3$ matrix. (a) Prove that the quadratic equation $\mathbf{x}^T K \mathbf{x}=1$ defines an ellipsoid in $\mathbb{R}^3$. What are its principal axes and semi-axes? (b) Describe the surface defined by the equation $11 x^2-8 x y+20 y^2-10 x z+8 y z+11 z^2=1$.

Let $K$ be a positive definite $3 \times 3$ matrix. (a) Prove that the quadratic equation $\mathbf{x}^T K \mathbf{x}=1$ defines an ellipsoid in $\mathbb{R}^3$. What are its principal axes and semi-axes?
(b) Describe the surface defined by the equation $11 x^2-8 x y+20 y^2-10 x z+8 y z+11 z^2=1$.

Key Concepts

Ellipsoids and Principal Axes

An ellipsoid in three dimensions is the set of points satisfying a scaled sum of squares equality. When expressed in a coordinate system aligned with the eigenvectors of its defining matrix, the equation reduces to the standard form of an ellipsoid. Here, the principal axes are the directions corresponding to these eigenvectors and the lengths of the semi-axes are determined by the inverses of the square roots of the eigenvalues.

Spectral Theorem and Diagonalization

The spectral theorem guarantees that every real symmetric matrix can be diagonalized by an orthogonal transformation. Diagonalization converts the quadratic form into a sum of squares of variables, each scaled by the corresponding eigenvalue. This process simplifies the analysis and classification of the surface defined by the quadratic form, by transforming it into a canonical form where the contributions along orthogonal directions (principal axes) are clearly separated.

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors provide critical insight into the structure of quadratic forms. In the context of ellipsoids defined by quadratic equations, the eigenvectors of the defining matrix give the directions of the ellipsoid’s principal axes, while the eigenvalues relate inversely to the squares of the lengths of the semi-axes. This relationship allows one to fully characterize the ellipsoid’s geometric properties.

Quadratic Forms

A quadratic form in n dimensions is an expression that can be written as x^T K x, where K is a symmetric matrix. This formulation encapsulates second-degree polynomial relationships in variables, and its properties can be analyzed using matrix algebra. In geometry, quadratic forms describe conic sections and quadric surfaces whose shapes are determined by the underlying matrix.

Positive Definite Matrices

A matrix is positive definite if it is symmetric and all its eigenvalues are positive, which implies that the quadratic form x^T K x is strictly positive for all nonzero vectors x. This property ensures that the level set defined by x^T K x = 1 is a closed, bounded surface, typically corresponding to an ellipsoid in three dimensions.

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