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 connections between and among women are the most feared, the most problematic, and the most potentially transforming force on the planet.”
—Adrienne Rich (b. 1929)
“... 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)
“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)