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The discriminant of a quadratic form, concretely the class of the determinant of a representing matrix in K / (K ×) 2 (up to non-zero squares) can also be defined, and for a real quadratic form is a cruder invariant than signature, taking values of only "positive, zero, or negative".
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
The signature of a metric tensor is defined as the signature of the corresponding quadratic form. [2] It is the number (v, p, r) of positive, negative and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities.
An alternative approach that uses the matrix form of the quadratic equation is based on the fact that when the center is the origin of the coordinate system, there are no linear terms in the equation. Any translation to a coordinate origin (x 0, y 0), using x* = x – x 0, y* = y − y 0 gives rise to
The pair (p, q) called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix. The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted O(p, q). Moreover, as a quadratic form and its opposite have the same orthogonal group ...
A symmetric real matrix A is called positive-definite if the associated quadratic form = has a positive value for every nonzero vector x in . If f ( x ) only yields negative values then A is negative-definite ; if f does produce both negative and positive values then A is indefinite . [ 30 ]
for a matrix = (,) with and running from 0 to . When the characteristic of the field of the coefficients is not two, generally , =, is ... the quadratic form ...
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.