Diagonalization of Quadratic Forms by Gauss Elimination

Abstract

The La Grange linear similarity transformation (completing the square) can be used to remove all cross-product terms from a quadratic form. It is shown that the La Grange transformation may be found conveniently by adapting the well-known Gauss elimination procedure for solving linear equations. A simple algorithm for finding the inverse transformation is given. This diagonalization scheme takes much less effort than finding the characteristic roots and vectors. It produces important simplifications in quadratic programming, statistics, and optimization problems.

Multivariate lag-windows and group representations

“…For instance, if h(x, y) = x + y or h(x, y) = x y, thenλ will be forced to be even. For k > 2, intricate symmetries quickly mount up as seen in (2) and the solution to the problem with general k will be addressed in the following sections.…”

“…is equivalent to requiring the entire string of equalities in (2), and in general, any set of equations is sufficient as long as the corresponding permutations generate all of S 3 . So we have established a three-way correspondence between elements of S 3 , matrices in GL 2 (R), and symmetries of C(x, y).…”

“…a diagonalization of the quadratic form (33) to standard form with rank and signature k − 1. This transformation is easily computed for any given k [2], and the Jacobian of the transformation can be used to determine α for a given β and k. For instance for k = 4, the substitutions…”

“…A methodical procedure is now presented that produces each of the symmetries of C(x, y) in (2). Under the commutativity property of multiplication, the product X t X t+x X t+y remains the same under any permutation of the three variables, so for each permutations in S 3 , we permute the variables, according to the permutation, then adjust t so that the first variable has index t. Then the symmetry condition can be read off from the last two variables.…”

Complex Static Skew-Symmetric Output Feedback Control

“…It is classical (e.g., proved by Gaussian elimination [1]) that any two invertible complex symmetric matrices A, B are (transpose) congruent: there exists an invertible matrix X such that X ⊤ AX = B. Similarly, any two invertible complex skew-symmetric matrices are congruent.…”