positievedefiniete

positievedefiniete refers to a property of certain mathematical objects, most notably square matrices and quadratic forms. A real symmetric matrix is called positive-definite if for any non-zero vector x, the quadratic form xᵀAx is strictly positive. Here, xᵀ denotes the transpose of vector x, and Ax is the matrix-vector product. This condition implies that all eigenvalues of the matrix A are strictly positive. Equivalently, a matrix is positive-definite if and only if all its leading principal minors are strictly positive.

In the context of quadratic forms, a function q(x) = xᵀAx is positive-definite if q(x) > 0 for all non-zero vectors x. This means the quadratic form never takes on a negative or zero value unless x is the zero vector. This property is crucial in various fields such as optimization, where it helps determine if a critical point is a minimum, and in statistics, particularly in the analysis of variance and covariance.

The concept can also be extended to complex Hermitian matrices, where the condition becomes xᴴAx > 0 for all non-zero complex vectors x, and xᴴ denotes the conjugate transpose. The notion of positive-definiteness is fundamental in understanding the behavior of systems described by these mathematical structures.