Féldefinitness

Féldefinitness is a concept in mathematics, specifically within the study of quadratic forms and matrices. A quadratic form is a polynomial of degree two in several variables. A matrix is said to be féldefinite if the quadratic form associated with it has a specific property related to its values.

There are several categories of féldefinitness. A quadratic form is positive féldefinite if it evaluates to zero for the zero vector and to a positive number for all non-zero vectors. Correspondingly, a matrix is positive féldefinite if it is symmetric and all its eigenvalues are positive.

A quadratic form is negative féldefinite if it evaluates to zero for the zero vector and to a negative number for all non-zero vectors. A symmetric matrix is negative féldefinite if all its eigenvalues are negative.

A quadratic form is non-negative féldefinite if it evaluates to zero for the zero vector and to a non-negative number for all vectors. A symmetric matrix is non-negative féldefinite if all its eigenvalues are non-negative.

Similarly, a quadratic form is non-positive féldefinite if it evaluates to zero for the zero vector and to a non-positive number for all vectors. A symmetric matrix is non-positive féldefinite if all its eigenvalues are non-positive.

A quadratic form is indefinite if it can take both positive and negative values. A symmetric matrix is indefinite if it has both positive and negative eigenvalues. The term "féldefinite" is often used as a general descriptor for these categories, with the specific type (positive, negative, non-negative, non-positive) clarifying the behavior of the quadratic form or the properties of the matrix. This concept is crucial in optimization problems, stability analysis, and various areas of applied mathematics and physics.