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An integral quadratic form has integer coefficients, such as x 2 + xy + y 2; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning Q(x, y) ∈ Z if x, y ∈ Λ.
Since the quadratic form is a scalar quantity, = (). Next, by the cyclic property of the trace operator, [ ()] = [ ()]. Since the trace operator is a linear combination of the components of the matrix, it therefore follows from the linearity of the expectation operator that
By definition, a quadric X of dimension n over a field k is the subspace of + defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables , …, +. (A homogeneous polynomial is also called a form, and so q may be called a quadratic form.)
In 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 .
The oldest problem in the theory of binary quadratic forms is the representation problem: describe the representations of a given number by a given quadratic form f. "Describe" can mean various things: give an algorithm to generate all representations, a closed formula for the number of representations, or even just determine whether any ...
The definition of a projective quadric in a real projective space (see above) can be formally adapted by defining a projective quadric in an n-dimensional projective space over a field. In order to omit dealing with coordinates, a projective quadric is usually defined by starting with a quadratic form on a vector space. [4]
The isotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces. [1] Over the real numbers, more generally in the case where F is a real closed field (so that the signature is defined), if the quadratic form is non-degenerate and has the signature (a, b), then its isotropy index is the minimum of a and b.
A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. [notes 2] A form of degree 2 is a quadratic form. In geometry, the Euclidean distance is the square root of a quadratic form.