title: Quadratic form (statistics)
text: In multivariate statistics, if  is a vector of  random variables, and  is an -dimensional symmetric matrix, then the scalar quantity  is known as a quadratic form in .
  
  Expectation
  It can be shown that
  
  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 that of Mathai and Provost.
  
  Proof 
  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
  
   
  
  A standard property of variances then tells us that this is
  
   
  
  Applying the cyclic property of the trace operator again, we get
  
  Variance in the Gaussian case
  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):
  
  .
  
  In addition, a quadratic form such as this follows a generalized chi-squared distribution.
  
  Computing the variance in the non-symmetric case
  Some texts incorrectly state that the above variance or covariance results hold without requiring  to be symmetric. The case for general  can be derived by noting that
  
  so
  
  is a quadratic form in the symmetric matrix , so the mean and variance expressions are the same, provided  is replaced by  therein.
  
  Examples of quadratic forms
  In the setting where one has a set of observations  and an operator matrix , then the  residual sum of squares can be written as a quadratic form in :
  
  For procedures where the matrix  is symmetric and idempotent, and the errors are Gaussian with covariance matrix ,  has a chi-squared distribution with  degrees of freedom and noncentrality parameter , where
  
  may be found by matching the first two central moments of a noncentral chi-squared random variable to the expressions given in the first two sections. If  estimates  with no bias, then the noncentrality  is zero and  follows a central chi-squared distribution.
  
  See also
  Quadratic form
  Covariance matrix
  Matrix representation of conic sections
  
  References 
  
  Statistical theory
  Quadratic forms