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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 sum of the first three terms of this equation, namely + + = (/ /) (), is the quadratic form associated with the equation, and the matrix = (/ /) is called the matrix of the quadratic form. The trace and determinant of A 33 {\displaystyle A_{33}} are both invariant with respect to rotation of axes and translation of the plane (movement of ...
For example, if A is a 3-by-0 matrix and B is a 0-by-3 matrix, then AB is the 3-by-3 zero matrix corresponding to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix. There is no common notation for empty matrices, but most computer algebra systems allow creating and computing with them.
As an example, let =, and consider the quadratic form = +where = [,] and c 1 and c 2 are constants. If c 1 > 0 and c 2 > 0 , the quadratic form Q is positive-definite, so Q evaluates to a positive number whenever [,] [,] .
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.
A general quadratic form on real variables , …, can always be written as where is the column vector with those variables, and is a symmetric real matrix. Therefore, the matrix being positive definite means that f {\displaystyle f} has a unique minimum (zero) when x {\displaystyle \mathbf {x} } is zero, and is strictly positive for any other x ...
If is the coefficient matrix of some quadratic form of , then is the matrix for the same form after the change of basis defined by . A symmetric matrix A {\displaystyle A} can always be transformed in this way into a diagonal matrix D {\displaystyle D} which has only entries 0 {\displaystyle 0} , + 1 {\displaystyle +1 ...
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