Let A ∈Mn, let 𝐱 ∈ℂ^n, and consider the complex quadratic form qA(𝐱)=𝐱^⊤ A 𝐱. (a) Show that qA(𝐱

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Let A ∈Mn, let 𝐱 ∈ℂ^n, and consider the complex quadratic...

Let $A \in \mathrm{M}_n$, let $\mathbf{x} \in \mathbb{C}^n$, and consider the complex quadratic form $q_A(\mathbf{x})=\mathbf{x}^{\top} A \mathbf{x}$. (a) Show that $q_A(\mathbf{x})=q_{\frac{1}{2}\left(A+A^{\top}\right)}(\mathbf{x})$, so we may assume that $A$ is symmetric. (b) If $A$ is symmetric and has singular values $\sigma_1, \sigma_2, \ldots, \sigma_n$, show that there is a change of variables $\mathbf{x} \mapsto U \mathbf{x}=\mathbf{y}$ $=\left[y_i\right]$ such that $U$ is unitary and $q_A(\mathbf{y})=\sigma_1 y_1^2+\sigma_2 y_2^2+\cdots+\sigma_n y_n^2$.

Key Concepts

Singular Values

Singular values are nonnegative real numbers associated with a matrix, representing the square roots of the eigenvalues of A?A. In many contexts, including diagonalization of quadratic forms, they provide a measure of the 'size' or 'stretching' effect of the matrix in different directions. When a symmetric matrix is diagonalized via a suitable change of variables, its representation in terms of singular values offers insight into its geometric action and scaling properties along specific directions.

Unitary Transformations

A unitary transformation is a linear change of variables represented by a unitary matrix, which preserves length and the inner product in complex vector spaces. Unitary matrices are the complex analogues of orthogonal matrices and are crucial in diagonalizing matrices and simplifying quadratic forms, especially when dealing with complex eigenvalues or singular values while maintaining the structure of the space.

Quadratic Forms

A quadratic form is a homogeneous polynomial of degree two in a set of variables. In linear algebra, quadratic forms are frequently written in matrix notation as x?Ax, where A is a matrix and x is a vector. They are central to many areas of mathematics, including optimization, geometry, and the study of conic sections, because they capture the second order structure of functions and allow for classification by definiteness, among other properties.

Matrix Symmetrization

Symmetrization is the process of converting an arbitrary square matrix into a symmetric matrix by averaging it with its transpose, i.e. (A+A?)/2. This process is important because many properties of quadratic forms depend only on the symmetric part of the matrix, and symmetric matrices have well?understood spectral properties that facilitate analysis and diagonalization.

Diagonalization of Symmetric Matrices

Diagonalization refers to the process of finding a change of basis in which a symmetric matrix is represented as a diagonal matrix. This is possible because symmetric matrices have a complete set of eigenvectors. In the context of quadratic forms, diagonalizing the matrix simplifies the form to a sum of terms each involving a squared variable, which greatly aids in understanding its behavior and classification.

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