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Transformers - 1984 The Key to Vector Sigma Part 1 2-39 is rated/received certificates of:

Australia:G

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How prove that variance-covariance matrix is nonnegative definite?

To prove that the variance-covariance matrix ( \Sigma ) is nonnegative definite, we can show that for any vector ( x ), the quadratic form ( x^T \Sigma x \geq 0 ). The variance-covariance matrix is defined as ( \Sigma = E[(X - E[X])(X - E[X])^T] ), where ( X ) is a random vector. By substituting ( x^T \Sigma x ) and using the properties of expected values and the definition of variance, we find that the expression equals the variance of the linear combination of the components of ( X ), which is always nonnegative. Thus, ( \Sigma ) is nonnegative definite.


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