sqrtqb

sqrtqb refers to the square root of a quadratic form, typically written as sqrt(qb(x)) where qb is a quadratic form on a vector x. In most common usage, qb is given by qb(x) = x^T Q x, with Q a symmetric real matrix. The function sqrtqb maps x to the nonnegative real number sqrt(x^T Q x).

If Q is positive semidefinite, qb(x) is always nonnegative, so sqrt(qb(x)) is defined for all x. If Q is positive definite, sqrtqb defines a norm on the Euclidean space R^n. In that case, the value is bounded by the eigenvalues of Q: for all x, sqrt(lambda_min) ||x|| ≤ sqrt(qb(x)) ≤ sqrt(lambda_max) ||x||, where lambda_min and lambda_max are the smallest and largest eigenvalues of Q. The gradient of sqrtqb at x ≠ 0 is ∇ sqrtqb(x) = Qx / sqrt(qb(x)).

sqrtqb is closely related to norms induced by quadratic forms and to ellipsoidal geometry. The set {x : qb(x) = c} is an ellipsoid in R^n when Q is positive definite, and sqrtqb(x) measures the distance from the origin in the Q-metric. In statistics and data analysis, special cases include the Mahalanobis norm, where Q corresponds to the inverse of a covariance matrix, yielding the ellipsoidal contours of multivariate normal distributions.

Examples: if Q = I, sqrtqb(x) = ||x||2, the standard Euclidean norm. If Q = diag(4, 1) in R^2, sqrtqb([x1, x2]) = sqrt(4 x1^2 + x2^2).

In practice, sqrtqb provides a way to quantify length or distance under a weighted quadratic form, with applications in optimization, geometry, and multivariate analysis.