Kantorovich inequality

The Kantorovich inequality bounds the product of quadratic forms of a positive-definite operator and its inverse by a quantity determined by the operator’s extreme eigenvalues.

Rayleigh quotient: A scalar formed by evaluating a quadratic form on a nonzero vector and dividing by the vector’s squared norm. The extreme eigenvalues bound the Rayleigh quotients that enter the inequality.
Conjugate gradient method: An iterative method for solving symmetric positive-definite linear systems. Kantorovich bounds help analyze progress and error in positive-definite system solving.
Leonid Kantorovich: A Soviet mathematician and economist whose work helped found linear programming and earned the 1975 Nobel Memorial Prize in Economic Sciences. The inequality bears his name and belongs to a broad tradition of his mathematical work.
Kantorovich constant: The sharp scalar factor [K=(M+m)^2/(4Mm)[0] associated with endpoints [m[0] and [M[0] of a positive interval. It is the exact factor that bounds the product in the normalized inequality.
Spectral theorem: A theorem that represents a self-adjoint operator using an orthonormal eigenbasis and real eigenvalues. Expanding the vector in an eigenbasis reduces the bound to eigenvalue extremes.
Matrix preconditioning: Transforming a linear system to improve its conditioning before numerical solution. The inequality connects spectral conditioning to bounds used when assessing preconditioners.
Kantorovich duality: A duality principle in optimization that relates a problem to a corresponding lower-bound problem. It shares Kantorovich’s name but concerns optimization duality, not this spectral bound.
Grüss inequality: An inequality bounding the deviation of an average product from the product of averages. Grüss bounds covariance-like deviations, whereas Kantorovich bounds paired quadratic forms.
Positive-definite matrix: A symmetric matrix whose quadratic form is strictly positive on every nonzero vector. The matrix form is a standard setting for the inequality and its inverse.
Steepest descent: An optimization method that repeatedly moves opposite the gradient of an objective function. Its convergence-rate estimates can be expressed through condition-number bounds of this kind.

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Popoviciu's inequalityRelated: It is a related sharp inequality whose bounds also exploit interval constraints.