Let 𝐯1, …, 𝐯n be elements of a complex inner product space....

Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be elements of a complex inner product space. Let $K$ denote the corresponding $n \times n$ Gram matrix, defined in the usual manner. (a) Prove that $K$ is a Hermitian matrix, as defined in Exercise 3.6.45. (b) Prove that $K$ is positive semi-definite, meaning $\mathbf{z}^T K \mathbf{z} \geq 0$ for all $\mathbf{z} \in \mathbb{C}^n$. (c) Prove that $K$ is positive definite if and only if $\mathbf{v}_1, \ldots, \mathbf{v}_n$ are linearly independent.

Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be elements of a complex inner product space. Let $K$ denote the corresponding $n \times n$ Gram matrix, defined in the usual manner.
(a) Prove that $K$ is a Hermitian matrix, as defined in Exercise 3.6.45.
(b) Prove that $K$ is positive semi-definite, meaning $\mathbf{z}^T K \mathbf{z} \geq 0$ for all $\mathbf{z} \in \mathbb{C}^n$.
(c) Prove that $K$ is positive definite if and only if $\mathbf{v}_1, \ldots, \mathbf{v}_n$ are linearly independent.

Key Concepts

Gram Matrix

A Gram matrix is constructed from a set of vectors in an inner product space by taking the inner product of each pair of vectors. The (i,j)-entry of a Gram matrix is defined as the inner product of the i-th and j-th vectors. This concept is fundamental in understanding how the angles and distances between vectors are interrelated, and it plays an important role in various areas such as numerical analysis, functional analysis, and machine learning.

Hermitian Matrix

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. In the context of a Gram matrix, the Hermitian property follows from the symmetry of the inner product, specifically because the inner product of two vectors is equal to the complex conjugate of the inner product taken in reverse order. This property ensures that many of the powerful spectral theorems applicable to real symmetric matrices extend to complex settings.

Positive Semi-Definite Matrix

A matrix is positive semi-definite if, for any complex vector, the associated quadratic form is non-negative. This property of Gram matrices originates from the definition of the inner product, which inherently preserves non-negativity. Positive semi-definiteness is crucial in many theoretical and applied contexts, including optimization problems and statistical analysis, providing assurances about the behavior of quadratic forms.

Positive Definite Matrix

A matrix is positive definite if its associated quadratic form is strictly positive for all non-zero vectors. In the context of Gram matrices, this property is intimately linked with the vectors from which the matrix is constructed; specifically, the Gram matrix is positive definite if and only if the set of vectors used to build it is linearly independent. Positive definiteness guarantees unique solutions in systems of equations and stability in optimization contexts.

Linear Independence

Linear independence is a fundamental concept in linear algebra which indicates that no vector in a set can be written as a linear combination of the others. In the context of Gram matrices, the property of positive definiteness directly correlates to the linear independence of the underlying vectors. The failure of linear independence, leading to redundancy among vectors, results in the Gram matrix being only positive semi-definite rather than positive definite.

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