Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form:
where a, ..., f are the coefficients. Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0).
The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. In the algebraic theory of quadratic forms, the coefficients are elements of a certain field. In the arithmetic theory of quadratic forms, the coefficients belong to a fixed commutative ring, frequently the integers Z or the p-adic integersZp. Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms. The theory of integral quadratic forms in n variables has important applications to algebraic topology.
where and are the expected value and variance-covariance matrix of , respectively, and tr denotes the trace of a matrix. This result only depends on the existence of and ; in particular, normality of is not required.
A book treatment of the topic of quadratic forms in random variables is
Proof
Since the quadratic form is a scalar quantity . Note that both and are linear operators, so . It follows that
and that, by the cyclic property of the trace operator,
Variance
In general, the variance of a quadratic form depends greatly on the distribution of . However, if does follow a multivariate normal distribution, the variance of the quadratic form becomes particularly tractable. Assume for the moment that is a symmetric matrix. Then,
In fact, this can be generalized to find the covariance between two quadratic forms on the same (once again, and must both be symmetric):
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is a 
is an 
is known as a quadratic form in 
and 
are the ![\operatorname{E}\left[\epsilon^T\Lambda\epsilon\right] = \operatorname{tr}(\operatorname{E}[\epsilon^T\Lambda\epsilon])](http://assets.wn.com/wiki/en/d/79/fe13f7ce2f3dcac3d0645-202567.png)
. Note that both 
and 
are linear operators, so 
. It follows that
and 
must both be symmetric):
Hastings Tribune
14 Dec 2022
Hastings Tribune
14 Dec 2022