GRAM DETERMINANT Meaning and
Definition
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The gram determinant, also known as the Gramian determinant, is a mathematical concept used in linear algebra. It is a determinant associated with the Gramian matrix, which is a square matrix created by multiplying the transpose of a given matrix by itself.
More specifically, given a set of vectors in a vector space, the Gramian matrix is formed by taking the inner product of each pair of vectors and assigning the result to the corresponding entry in the matrix. The resulting matrix is symmetric and positive semi-definite.
The gram determinant refers to the determinant of the Gramian matrix. It is calculated by evaluating the determinant of the matrix. This determinant serves as a measure of linear independence or dependence of the vectors involved. If the gram determinant is non-zero, it indicates that the vectors are linearly independent, and if the determinant is zero, it indicates that the vectors are linearly dependent.
In addition to determining linear independence, the gram determinant has numerous applications in various fields. It is widely used in signal processing, statistics, and numerical methods. For example, in signal processing, the gram determinant can be used to assess the effectiveness of different sets of basis signals and to quantify the amount of information contained in a particular set of signals. In statistics, it can be used to assess the multicollinearity, or linear dependence, of predictor variables in regression analysis. Overall, the gram determinant provides valuable insights into vector relationships and is an important tool in linear algebra and related disciplines.
Etymology of GRAM DETERMINANT
The word "gram determinant" has its roots in mathematics, specifically linear algebra.
The term "gram" comes from the German mathematician Jørgen Pedersen Gram (1850-1916), who made significant contributions to the theory of determinants and matrices. As a result, the term "gram" is often used to describe mathematical concepts related to determinants and matrices.
The word "determinant" has its origins in Latin. "Determinans" is the present participle of the Latin verb "determinare", meaning "to determine". In mathematics, a determinant is a value associated with a square matrix that provides important information about its properties and transformations.
So, the term "gram determinant" refers to a determinant conceptually associated with the work of Jørgen Pedersen Gram. The gram determinant is specifically used to determine whether a set of vectors in a vector space is linearly independent or not.
