Let · be any norm on ℝ^n. (a) Show that q(𝐱) is a positive definite quadratic form if and only i

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Let · be any norm on ℝ^n. (a) Show that q(𝐱) is a positive...

Let $\|\cdot\|$ be any norm on $\mathbb{R}^n$. (a) Show that $q(\mathbf{x})$ is a positive definite quadratic form if and only if $q(\mathbf{u})>0$ for all unit vectors, $\|\mathbf{u}\|=1$. (b) Prove that if $S=S^T$ is any symmetric matrix, then $K=S+c$ I $>0$ is positive definite if $c$ is sufficiently large.

Key Concepts

Positive Definite Quadratic Forms

A quadratic form is considered positive definite if it produces only positive values when evaluated at any nonzero vector, and it equals zero only when the vector is the zero vector. This property ensures that the quadratic form has a strict notion of 'positivity' which is fundamental in optimization and stability analysis, and it is often characterized in terms of its associated matrix's eigenvalues being all positive.

Unit Vectors in Normed Spaces

The concept of unit vectors, which are vectors with norm equal to one, is used to simplify the analysis of vector spaces by restricting attention to a compact set. In the context of quadratic forms, checking the sign on all unit vectors is sufficient because any nonzero vector can be scaled to a unit vector, making this approach both necessary and efficient for verifying positive definiteness.

Symmetric Matrices and Eigenvalues

Symmetric matrices are central in the study of quadratic forms because their eigenvalues are real, which provides a clear criterion for positive definiteness. The spectral theorem guarantees that a symmetric matrix can be diagonalized via an orthogonal transformation, making the relationship between the quadratic form and the eigenvalues explicit—specifically, a symmetric matrix yields a positive definite quadratic form if and only if all its eigenvalues are positive.

Eigenvalue Shifting

Adding a scalar multiple of the identity matrix to a symmetric matrix results in shifting all the eigenvalues by that scalar. This technique, known as eigenvalue shifting, is particularly useful because it provides a method to ensure that a matrix becomes positive definite. By choosing a sufficiently large constant, even a matrix that is not positive definite can be adjusted so that all of its eigenvalues, and hence the corresponding quadratic form, are positive.

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