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:
“I have no connections here; only gusty collisions,
rootless seedlings forced into bloom, that collapse.
...
I am the Visiting Poet: a real unicorn,
a wind-up plush dodo, a wax museum of the Movement.
People want to push the buttons and see me glow.”
—Marge Piercy (b. 1936)
“The quickness with which all the “stuff” from childhood can reduce adult siblings to kids again underscores the strong and complex connections between brothers and sisters.... It doesn’t seem to matter how much time has elapsed or how far we’ve traveled. Our brothers and sisters bring us face to face with our former selves and remind us how intricately bound up we are in each other’s lives.”
—Jane Mersky Leder (20th century)
“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)