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:
“The conclusion suggested by these arguments might be called the paradox of theorizing. It asserts that if the terms and the general principles of a scientific theory serve their purpose, i. e., if they establish the definite connections among observable phenomena, then they can be dispensed with since any chain of laws and interpretive statements establishing such a connection should then be replaceable by a law which directly links observational antecedents to observational consequents.”
—C.G. (Carl Gustav)
“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)
“Imagination is an almost divine faculty which, without recourse to any philosophical method, immediately perceives everything: the secret and intimate connections between things, correspondences and analogies.”
—Charles Baudelaire (1821–1867)