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Quadratic Forms
Published in Ravi P. Agarwal, Cristina Flaut, An Introduction to Linear Algebra, 2017
Ravi P. Agarwal, Cristina Flaut
Quadratic forms occur naturally in physics, economics, engineering (control theory), and analytic geometry (quadratic curves and surfaces). Particularly, recall that the equation (quadratic form) of a central quadratic curve in a plane, after translating the origin of the rectangular coordinate system to the centre of the curve, appears as q2(x,y)=(x,y)abbcxy=ax2+2bxy+cy2=d. $$ q_{2} (x,~y) = (x,~y)\left( {\begin{array}{*{20}c} a & b \\ b & c \\ \end{array} } \right)\left( {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right)~ = ax^{2} + 2bxy + cy^{2} = d. $$
Geometry of the Middle Surface
Published in Eduard Ventsel, Theodor Krauthammer, Thin Plates and Shells, 2001
Eduard Ventsel, Theodor Krauthammer
The first quadratic form generally pertains to the measurement of distances along an arc between two points on a surface, to areas on the surface, and to angles between any two curves passing through a point (i.e., the angle between the tangents to these curves), etc. Thus, the first quadratic form defines the intrinsic geometry of the surface. However, it does not involve the specific shape of the surface. For instance, this form cannot be a measure of curvature of curves lying on a surface or if we change a curvature of the surface (say, by its bending) without its stretching or contracting, the first quadratic form remains unchanged. So, the representation of the surface by the Lamé parameters only is not sufficient and the first quadratic form must be complemented with the second quadratic form to describe uniquely the intrinsic and extrinsic geometries of a surface.
Orthogonality
Published in Crista Arangala, Exploring Linear Algebra, 2019
A quadratic form is a function on Rn where QA(x) = xTAx, or QA(x1,x2,x3,⋯,xn)=∑i≤jaijxixj, and A is a symmetric matrix. The matrix A is called the matrix for the form. Note QA(0→)=0. For example QA: R2 → R2 defined by QA(x1,x2)=x12+2x1x2+x22 is a quadratic form.
Finite volume element methods for two-dimensional time fractional reaction–diffusion equations on triangular grids
Published in Applicable Analysis, 2023
Zhichao Fang, Jie Zhao, Hong Li, Yang Liu
Making use of Lemma 3.1, we can easily obtain that the matrix is symmetric positive definite. Let , then (33) can be rewritten as follows Next, we will prove G is invertible. Applying Lemma 3.3, for , we have where . This means that (for ) is a positive definite quadratic form generated by the nonsymmetric matrix . Therefore, (for ) is a positive definite quadratic form generated by the nonsymmetric matrix G, then we have that G is invertible. In fact, if G is noninvertible, then the homogeneous linear equations GY = 0 has nonzero solution , thus, we have which is in contradiction with the definition of positive definite quadratic form. Hence, G is invertible, then the linear Equation (33) have a unique solution. This shows that the fully discrete FVE scheme (11) has a unique solution. Then, we complete the proof.
On the Sturm–Liouville problem describing an ocean waveguide covered by pack ice
Published in Applicable Analysis, 2022
Boris P. Belinskiy, Don B. Hinton, Lakmali Weerasena, Mohammad M. Khan
We denote the norm by Since , it follows from this theorem that and realizes this minimum. Since we conclude Choose and define the quadratic form K by Then K is a positive definite quadratic form. The set is then given an inner product and norm by Also for , and using the orthogonality of the in and definition (29) Hence, are orthogonal in
An SOS method for the design of continuous and discontinuous differentiators
Published in International Journal of Control, 2018
Tonametl Sanchez, Emmanuel Cruz-Zavala, Jaime A. Moreno
Some important characteristics of Algorithm 5.1 are the following: The algorithm is always feasible in the sense that for every valid values of n, d, p and β, there exists a gain vector rendering the GF W(z) positive definite, and for a large enough q in Theorem 5.1, the forms are SOS. This important feature, due to Theorems 3.1 and 5.1, assures that the search is not blind and the number of iterations is finite.The problem is linear in .It is possible to iterate the algorithm by changing β, i.e. going from (6) to (3), finding different values of k. The problem is bilinear in .Since any positive-definite quadratic form is SOS, see. e.g. Marshall (2008), the algorithm is always feasible for linear differentiators with quadratic LFs, i.e. Algorithm 5.1 is always feasible for d = 0, p = 2, for any n.