SEMIDEFINITE Meaning and
Definition
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The term "semidefinite" refers to a mathematical concept that is used to describe a specific type of matrix known as a semidefinite matrix. In linear algebra, a semidefinite matrix is a square matrix that has the property of being positive semidefinite.
Positive semidefiniteness refers to a property of a matrix where all of its eigenvalues are either non-negative or zero. An eigenvalue is a scalar value that is associated with a specific vector, and it represents the magnitude of the vector when it is transformed by the matrix.
In simpler terms, a semidefinite matrix is a square matrix that satisfies the condition that all of its eigenvalues are greater than or equal to zero. This means that the matrix is not invertible, as it has at least one eigenvalue equal to zero.
Semidefinite matrices have various applications in fields such as optimization, control theory, signal processing, and even quantum mechanics. They are particularly useful in describing and solving optimization problems with linear constraints. Additionally, semidefinite programming is a powerful tool that utilizes semidefinite matrices to solve problems with objective functions that are linear in terms of semidefinite constraints.
Etymology of SEMIDEFINITE
The word "semidefinite" is derived from two separate terms: "semi-" and "definite".
1. "Semi-" is a prefix that means "half" or "partially". It comes from the Latin word "semi", which has the same meaning.
2. "Definite" comes from the Latin word "definitus", which means "defined" or "determined". It is derived from the verb "definire", meaning "to set limits" or "to define".
When combined, the term "semidefinite" suggests something that is partially or halfway determined, indicating a less restrictive or partial form of definiteness. In mathematics, particularly in linear algebra and optimization, the term "semidefinite" is commonly used to describe a particular type of matrix, called a semidefinite matrix, which has a kind of positive definiteness property but allows for zero eigenvalues.
