Understanding Positive Definiteness in Real and Complex Matrices

When dealing with matrices in mathematics and linear algebra, the concept of positive definiteness is an important property to understand, especially in the context of real and complex matrices.

Can a Non-Symmetric Real Matrix Be Positive Definite?

No, a non-symmetric real matrix cannot be positive definite. To understand why, let's first define what a positive definite matrix is. A matrix ( A ) is defined as positive definite if for all non-zero vectors ( x ), the following condition holds:

( x^T A x > 0 ) for all ( x eq 0 )

This definition hinges on the eigenvalues of the matrix. For a matrix to be positive definite, it must have all positive eigenvalues. However, non-symmetric matrices can have complex eigenvalues or may fail to meet the conditions for positive definiteness. The lack of symmetry can lead to situations where the quadratic form ( x^T A x ) does not always yield a positive value for all non-zero ( x ).

Inner Product-based Definitions

The concept of positive definiteness can be defined using different inner products, either real or complex.

Real Inner Product Definition

For a matrix ( A ) with real entries, ( A ) is positive definite if ( x^T A x geq 0 ) for each nonzero vector ( x ). Moreover, for the condition to strictly hold for positive definiteness, ( x^T A x > 0 ) must be true for all ( x eq 0 ). It is also noted that a matrix ( A ) is positive definite if and only if its symmetric part ( frac{1}{2}(A A^T) ) is positive definite.

Complex Inner Product Definition

In the case of complex matrices, the inner product is defined using the complex conjugate transpose, denoted as ( x^H ). A matrix ( A ) is positive definite if ( x^H A x geq 0 ) for each nonzero complex vector ( x ), and strictly ( x^H A x > 0 ) for all ( x eq 0 ).

A significant difference with the real case is that a positive definite matrix in the complex domain must necessarily be Hermitian, i.e., ( A^H A ). This means that the matrix is unchanged when the transpose and complex conjugate are applied. This requirement is due to the definition of positive definiteness in the complex case, which stipulates that the quadratic form ( x Ax ) is both real-valued and positive.

The Underlying Reason

The distinction arises from the nature of the real and complex cases. In the real case, the positive definiteness only requires that the quadratic form is strictly positive. However, in the complex case, there is an additional requirement that the form is real-valued. The real-valued condition is what necessitates the matrix being symmetric (Hermitian in the complex case).

Conclusion

Summarizing, only symmetric (or Hermitian in the complex case) matrices can be classified as positive definite. Their eigenvalues are real and can be evaluated to meet the positive definiteness criteria. The difference in definitions between real and complex matrices reflects the underlying mathematical properties and the nature of the inner products used.

Key Takeaways

A matrix is positive definite if ( x^T A x > 0 ) for all nonzero vectors ( x ) in the real case. In the complex case, a matrix is positive definite if ( x^H A x > 0 ) for all nonzero complex vectors ( x ). Only symmetric (or Hermitian) matrices can be positive definite due to the requirements for the quadratic form.

Understanding these distinctions is crucial for applications in linear algebra, optimization, and various areas of engineering and physics.