In economic optimization problems, these quantities represent shadow prices, and the requirement that these quantities be positive is one of the Kuhn-Tucker conditions.
In principal component analysis, the eigenvalues of the covariance matrix correspond to these quantities. In economic optimization problems, these quantities represent shadow prices, and the requirement that these quantities be positive is one of the Kuhn-Tucker conditions. In a dynamical system, the reaction forces are given by the sum over the derivatives with respect to generalized coordinates of the constraint function times these quantities. These quantities arise from the fact that at the extremum of a function's value as it moves along a curve, both that function's gradient and the curve gradient are parallel, and thus proportional to each other. For 10 points, identify these quantities, constants of proportionality that arise when a function is to be extremized under some specified constraints.
In principal component analysis, the eigenvalues of the covariance matrix correspond to these quantities.
Lagrange multipliers