Understanding Positive Definiteness: When Eigenvalues and Symmetry Matter

Understanding the concept of positive definiteness in matrices is crucial for various applications in mathematics, physics, and engineering. This article aims to provide a clear explanation of why a matrix is not considered positive definite under certain conditions, including the role of eigenvalues and symmetry.

Why Positive Definiteness is Defined for Symmetric Matrices

Positive definiteness is a property that only makes sense for symmetric matrices (or Hermitian matrices in the complex case). This is because the definition of positive definiteness is closely tied to the idea of quadratic forms, which are intimately connected with symmetric matrices. In the context of symmetric matrices, a non-symmetric matrix can be transformed into a symmetric one, making the concept of positive definiteness well-defined.

Conditions for Positive Definiteness

Let us delve into the conditions that determine whether a matrix is positive definite.

1. Symmetric Matrix with Positive Eigenvalues:

A symmetric matrix with all its eigenvalues being positive is positive definite. This means that for any non-zero vector (vec{x}), the quadratic form (vec{x}^T A vec{x} > 0).

2. Symmetric Matrix with Negative Eigenvalues:

If a symmetric matrix has any negative eigenvalues, it is not positive definite. In fact, it would be negative definite if all eigenvalues are negative and indefinite if the eigenvalues have mixed signs (some positive and some negative).

3. Symmetric Matrix with Some Zero Eigenvalues:

A symmetric matrix with some zero eigenvalues is positive (or negative) semidefinite if all the remaining eigenvalues are positive (or negative).

Relevance of Symmetry and Eigenvalues in Definiteness

The importance of symmetry and eigenvalues in the context of positive definiteness can be illustrated through the following example. Consider a matrix (A) with a negative eigenvalue (lambda), and let (vec{x}) be a corresponding eigenvector. Then:

[left( A vec{x} right) cdot left( vec{x} right) lambda left( vec{x} cdot vec{x} right)

This clearly shows a failure of the matrix (A) to be positive definite, as the quadratic form should be greater than zero for a positive definite matrix.

Necessity of Symmetry and Positive Eigenvalues

For a matrix to be considered positive definite, it must meet two key conditions:

Hermitian (or Symmetric): This condition ensures that the matrix is self-adjoint, which is essential for the definition of positive definiteness. All Positive Eigenvalues: The eigenvalues must all be positive, ensuring that the quadratic form is always positive.

The combination of these two conditions ensures that the matrix is positive definite:

Not Hermitian/Not Symmetric: If a matrix is not Hermitian or symmetric, it cannot be positive definite. Negative Eigenvalues: A matrix with negative eigenvalues cannot be positive definite, regardless of whether it is symmetric or not.

Conclusion and Proofs of Non-Definiteness

To prove that a matrix is not semi-definite (positive or negative), it is sufficient to demonstrate either that the matrix is not Hermitian or that it has non-positive eigenvalues. This ensures that the matrix fails to meet the necessary conditions for positive definiteness.

Understanding these conditions and their implications is crucial for various applications, such as optimization, spectral analysis, and solving linear systems. By ensuring that matrices meet the requirements for positive definiteness, we can guarantee that certain mathematical properties hold true, leading to reliable and accurate results in these applications.