puolidefiniittinen

puolidefiniittinen is a term used in mathematics, particularly in linear algebra and functional analysis, to describe a property of matrices and quadratic forms. A quadratic form associated with a real symmetric matrix A is said to be puolidefiniittinen if it takes non-negative values for all possible non-zero vectors x, and takes the value zero for at least one non-zero vector x. This is a refinement of the concept of positive semi-definiteness, which only requires the form to be non-negative.

In simpler terms, for a puolidefiniittinen quadratic form, the result of plugging in any non-zero vector is always greater than or equal to zero. Furthermore, there must be at least one specific non-zero vector that, when plugged into the form, yields exactly zero. This distinction is crucial in various areas, including optimization problems and the study of differential equations.

The matrix A itself is referred to as puolidefiniittinen if its corresponding quadratic form has this property. This means that x^T A x >= 0 for all vectors x, and there exists a non-zero vector y such that y^T A y = 0. This condition is met if and only if all eigenvalues of the real symmetric matrix A are non-negative, and at least one eigenvalue is zero. Therefore, a puolidefiniittinen matrix is a positive semi-definite matrix that is not positive definite.

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