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
Famous quotes containing the word connections:
“A foreign minister, I will maintain it, can never be a good man of business if he is not an agreeable man of pleasure too. Half his business is done by the help of his pleasures: his views are carried on, and perhaps best, and most unsuspectedly, at balls, suppers, assemblies, and parties of pleasure; by intrigues with women, and connections insensibly formed with men, at those unguarded hours of amusement.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“... feminism is a political term and it must be recognized as such: it is political in women’s terms. What are these terms? Essentially it means making connections: between personal power and economic power, between domestic oppression and labor exploitation, between plants and chemicals, feelings and theories; it means making connections between our inside worlds and the outside world.”
—Anica Vesel Mander, U.S. author and feminist, and Anne Kent Rush (b. 1945)
“Growing up human is uniquely a matter of social relations rather than biology. What we learn from connections within the family takes the place of instincts that program the behavior of animals; which raises the question, how good are these connections?”
—Elizabeth Janeway (b. 1913)