How Can the Variance of a Quadratic Form Be Simplified?

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SUMMARY

The variance of a quadratic form, as detailed in Searle's 1971 book "Linear Model," is expressed by the formula var(Y^{T}AY) = 2tr(AΣAΣ) + 4μ^{T}AΣAμ. This formula applies to a constant matrix A and a jointly Gaussian random vector Y. The discussion seeks simpler proofs for this variance calculation, particularly those that resemble the derivation of the expectation of a quadratic form. Participants suggest using a double sum approach for simplification.

PREREQUISITES
  • Understanding of quadratic forms in linear algebra
  • Familiarity with multivariate Gaussian distributions
  • Knowledge of matrix trace operations
  • Basic concepts of moment-generating functions (MGFs)
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  • Research alternative proofs for the variance of quadratic forms
  • Explore the derivation of the expectation of quadratic forms
  • Study the properties of moment-generating functions in relation to quadratic forms
  • Learn about the application of double sums in simplifying mathematical expressions
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Statisticians, mathematicians, and data scientists interested in simplifying variance calculations for quadratic forms and those working with multivariate Gaussian distributions.

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In the Searle's 1971 book Linear Model, page 57, has a formula for the Variance of Quadratic form:

var(Y^{T}AY)=2tr(A\SigmaA\Sigma)+4\mu^{T}A\SigmaA\mu

The proof of this showed on page 55 was based on MGF. I'm looking for proofs are less complicated. Some thing that is similar to show the expectation of a quadratic form.

Anyone has read about quadratic form please help.

Thanks
 
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If you're talking about a constant matrix A and a random vector Y that is jointly gaussian, one way is to write the quadratic form as a double sum.
 

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