Connections
The general purely quadratic real function f(z) on n real variables can always be written as zTM z where z is the column vector with those variables, and M is a symmetric real matrix. Therefore, the matrix being positive definite means that f has a unique minimum (zero) when z is zero, and is strictly positive for any other z.
More generally, a twice-differentiable real function f on n real variables has an isolated local minimum at arguments if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive definite at that point. Similar statements can be made for negative definite and semi-definite matrices.
In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear combination of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.
Read more about this topic: Positive-definite Matrix
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