Definite quadratic formIn mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.
Covariance matrixIn probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions.
Frobenius normal formIn linear algebra, the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation by invertible matrices over F. The form reflects a minimal decomposition of the vector space into subspaces that are cyclic for A (i.e., spanned by some vector and its repeated images under A). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix B is similar to A if and only if it has the same rational canonical form as A.
Null vectorIn mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space.
Sonata formSonata form (also sonata-allegro form or first movement form) is a musical structure generally consisting of three main sections: an exposition, a development, and a recapitulation. It has been used widely since the middle of the 18th century (the early Classical period). While it is typically used in the first movement of multi-movement pieces, it is sometimes used in subsequent movements as well—particularly the final movement.
Binary formBinary form is a musical form in 2 related sections, both of which are usually repeated. Binary is also a structure used to choreograph dance. In music this is usually performed as A-A-B-B. Binary form was popular during the Baroque period, often used to structure movements of keyboard sonatas. It was also used for short, one-movement works. Around the middle of the 18th century, the form largely fell from use as the principal design of entire movements as sonata form and organic development gained prominence.
Musical formIn music, form refers to the structure of a musical composition or performance. In his book, Worlds of Music, Jeff Todd Titon suggests that a number of organizational elements may determine the formal structure of a piece of music, such as "the arrangement of musical units of rhythm, melody, and/or harmony that show repetition or variation, the arrangement of the instruments (as in the order of solos in a jazz or bluegrass performance), or the way a symphonic piece is orchestrated", among other factors.
Isotropic lineIn the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form. Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point (α, β) that depend on the imaginary unit i: First system: Second system: Laguerre then interpreted these lines as geodesics: An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero.
Scaling (geometry)In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions. The result of uniform scaling is similar (in the geometric sense) to the original. A scale factor of 1 is normally allowed, so that congruent shapes are also classed as similar. Uniform scaling happens, for example, when enlarging or reducing a photograph, or when creating a scale model of a building, car, airplane, etc.
Diagonalizable matrixIn linear algebra, a square matrix is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix and a diagonal matrix such that , or equivalently . (Such , are not unique.) For a finite-dimensional vector space , a linear map is called diagonalizable if there exists an ordered basis of consisting of eigenvectors of .
