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Topic: Quadratic form

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Symmetric bilinear form

Quadratic function

Quadratic equation

Module homomorphism

Homogeneous (mathematics)

Commutative ring

Polynomial

Monomial

Chebyshev polynomials

Symmetric matrix

Diophantine approximation

Pythagorean theorem

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	PlanetMath: quadratic form
		The definition of equivalent quadratic forms is well-defined and it is not hard to see that this equivalence is an equivalence relation.
		The definiteness of a quadratic form is preserved under the equivalence relation on quadratic forms.
		This is version 39 of quadratic form, born on 2002-02-13, modified 2007-04-15.
	planetmath.org /encyclopedia/QuadraticForm.html (364 words)

	Quadratic equation - Wikipedia, the free encyclopedia
		In mathematics, a quadratic equation is a polynomial equation of the second degree.
		This equation may be resolved directly or with a simple substitution, using the methods that are available for the quadratic, such as factoring (also called factorising), the quadratic formula, or completing the square.
		The quadratic formula is derived by the method of completing the square.
	en.wikipedia.org /wiki/Quadratic_equation (871 words)

	Info and facts on 'Quadratic form' (Site not responding. Last check: 2007-10-21)
		Note that general quadratic function (additional info and facts about quadratic function) s and quadratic equation (An equation in which the highest power of an unknown quantity is a square) s are not examples of quadratic forms.
		Quadratic forms over the ring of integers are called integral quadratic forms or integral lattice (Framework consisting of an ornamental design made of strips of wood or metal) s.
		The kernel of the bilinear form B consists of the elements that are orthogonal to all elements of V, and the kernel of the quadratic form Q consists of all elements u of the kernel of B with Q(u)=0.
	www.absoluteastronomy.com /encyclopedia/q/qu/quadratic_form.htm (841 words)

	PlanetMath: diagonal quadratic form
		are all 0 in a diagonal quadratic form.
		Every quadratic form is equivalent to a diagonal quadratic form.
		This is version 9 of diagonal quadratic form, born on 2006-02-21, modified 2006-10-11.
	www.planetmath.org /encyclopedia/DiagonalQuadraticForm.html (128 words)

	[No title]
		Quadratic subspaces are subspaces on which this quadratic form is still non degenerate; completely singular (or isotropic) subspaces are subspaces on which this quadratic form vanishes.
		The quadratic space V is called neutral if it is the direct sum of two completely singular subspaces; this condition is equivalent to the existence of at least one completely singular subspace of dimension m (the half of dim V).
		A quadratic plane is either neutral or anisotropic (this means that 0 is the only singular (or isotropic) vector); and every neutral quadratic space is an orthogonal direct sum of neutral planes; moreover if V is neutral, the center of Cl_0 is isomorphic to K^2; these three assertions are true for fields of any characteristic.
	www.math.niu.edu /~rusin/known-math/99/clifford_alg (1678 words)

	PlanetMath: diagonalization of quadratic form
		A quadratic form may be diagonalized by the following procedure:
		The quadratic form is now diagonal, so we are done.
		This is version 4 of diagonalization of quadratic form, born on 2004-11-18, modified 2006-10-05.
	planetmath.org /encyclopedia/DiagonalizationOfQuadraticForm.html (161 words)

	Quadratic Functions
		A function may be transformed from quadratic form into standard form by completing the square.
		Regardless of the form in which the equation is expressed, the vertex is unique.
		A quadratic function may be expressed in two ways: quadratic form or standard form.
	jwilson.coe.uga.edu /emt668/EMT668.Folders.F97/Wynne/Quadratic/quadratic.html (961 words)

	Quadratic form
		In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.
		To express the quadratic form concept in linear algebra terms, we can note that for any bilinear form B on a vector space V of finite dimension, the expression B(v,v) for v in V will be a quadratic form in the co-ordinates of v with respect to a fixed basis.
		For the purposes of quadratic form theory over rings in general, such as the integral quadratic forms important in number theory and topology, one must start with a more careful definition to avoid problems caused by division by 2.
	www.brainyencyclopedia.com /encyclopedia/q/qu/quadratic_form.html (282 words)

	Math Help - Algebra - Quadratics - Theory - Graphs
		There are three main methods of solving quadratics: Guessing the solutions (also known as the double parentheses method), completing the square, and using the quadratic formula.
		There is a special form of a quadratic that is best for graphing the equation.
		Please note that we are ignoring the standard form for completing the square, since it is pretty easy to get this once the quadratic form is known.
	www.hyper-ad.com /tutoring/math/algebra/Quadratic_theory.html (1403 words)

	College Algebra Tutorial on Equations that are Quadratic in Form
		equal to 0, you have an equation that is quadratic in form.
		If it is not in standard form, move any term(s) to the appropriate side by using the addition/subtraction property of equality.
		Even though not all of the quadratic in form equations can cause extraneous solutions, it is better to be safe than sorry and just check them all.
	www.wtamu.edu /academic/anns/mps/math/mathlab/col_algebra/col_alg_tut20_quadform.htm (1230 words)

	Quadratic Form
		The quadratic form of a matrix M and a vector X is defined as:
		Quadratic forms are common in statistics, particularly in linear models and multivariate analysis.
		In Dataplot applications, the QUADRATIC FORM command is most typically used as an intermediate calculation in a larger macro.
	www.itl.nist.gov /div898/software/dataplot/refman2/auxillar/quadform.htm (277 words)

	Factoring Quadratic Equations
		If the left-hand side of the general form of a quadratic equation can be factored, the only way for the factored equation to be true is for one or both of the factors to be zero.
		Thus, the roots of quadratic equations which can be factored can be found by setting each of the factors equal to zero and solving the resulting linear equations.
		Quadratic equations in which the numerical constant c is zero can always be solved by factoring. One of the two roots is zero.
	www.tpub.com /doemathematics/mathematics46.htm (644 words)

	solving quadratic equations 6
		The factored form of a quadratic equation also tells you where the x-intercepts (sometimes called the roots or zeros of the quadratic function are located).
		The factored form must have a factor of (x - 1) to ensure that when you plug in x = 1 the value of y will be equal to zero.
		The factored form must also have a factor of (x - 4) to ensure that when you plug in x = 4 the value of y will be equal to zero.
	algebra-tutoring.com /solving-quadratic-equations-6.htm (685 words)

	Rotation
		Quadratic surfaces are here, under linear algebra, because the complete analysis of an arbitrary quadratic form involves matrices, eigen vectors, and eigen values.
		Run the quadratic form through q, and the matrix is diagonal, and all the mixed terms go away.
		The result is a quadratic form consisting of squared terms, a constant, and at most one linear term.
	www.mathreference.com /la-qf,rot.html (1230 words)

	Lecture Notes 19 - Math 4220
		The term quadratic refers to the fact that the coordinates of the points in these structures satisfy quadratic equations.
		The quadric determined by this quadratic form consists of the points with coordinates (1,2,0), (1,3,0), (2,0,1), (0,2,1), (3,0,1), and (0,3,1).
		When the field has characteristic 2 we can not use the quadratic formula, but alternate means of solving the quadratic equation exist and we get results that are the same as those in odd characteristic fields.
	www-math.cudenver.edu /~wcherowi/courses/m4220/hg2lec19.html (916 words)

	Background, McMath, Trident (Site not responding. Last check: 2007-10-21)
		The result of composition is another quadratic form similarly related to somewhere else in the continued fraction expansion.
		However, this algorithm also sometimes reverses the direction of the quadratic form, a trait that is undesirable for this research for reasons that will become apparent later, so quadratic forms will instead be converted back into a step in the continued fraction expansion in order to reduce them.
		The number of terms skipped is larger for a quadratic form farther down in the expansion and can be roughly approximated, so that if it is known which step in the process is desired, composition of quadratic forms gets close enough that the answer can be quickly found.
	web.usna.navy.mil /~wdj/mcmath/node2.html (1388 words)

	Algebra II: Quadratic Equations - Text-only
		Quadratic equations, or equations of the second degree, such as x^2 + 2x - 5 are probably the most common equation you will see in Algebra II (intermediate algebra).
		Any equation of type ax^2 + bx + c = 0 where a, b, and c are constants and a <> 0, is in standard form for a quadratic equation.
		Quadratic equations of type ax^2 + bx + c = 0 and ax^2 + bx = 0 (c is 0) can be factored to solve for x.
	library.advanced.org /20991/textonly/alg2/quad.html (591 words)

	Quadratic forms: conditions for semidefiniteness
		As in the case of two variables, to determine whether a quadratic form is positive or negative semidefinite we need to check more conditions than we do in order to check whether it is positive or negative definite.
		In particular, it is not true that a quadratic form is positive or negative semidefinite if the inequalities in the conditions for positive or negative definiteness are satisfied weakly.
		By studying quadratic forms we also study quadratic functions: by changing the variables we can reduce any quadratic function to a quadratic form.
	www.chass.utoronto.ca /~osborne/MathTutorial/QFS.HTM (954 words)

	Operations on Forms
		For a binary quadratic form f, returns its order as an element of the class group Cl(Q) where Q is the parent of f.
		In addition to the Parent and Category structures of binary quadratic forms of discriminant D, the quadratic forms map to the ideals of a fixed order or discriminant D in a quadratic number field.
		Given a structure of quadratic forms of discriminant D, returns the associated order of discriminant D in a quadratic field.
	www.umich.edu /~gpcc/scs/magma/text824.htm (476 words)

	Quadratic forms: conditions for semidefiniteness
		As in the case of two variables, to determine whether a quadratic form is positive or negative semidefinite we need to check more conditions than we do in order to check whether it is positive or negative definite.
		In particular, it is not true that a quadratic form is positive or negative semidefinite if the inequalities in the conditions for positive or negative definiteness are satisfied weakly.
		By studying quadratic forms we also study quadratic functions: by changing the variables we can reduce any quadratic function to a quadratic form.
	free.prohosting.com /cepr/data/adveco/qfs.html (895 words)

	BioMath: Quadratic Functions
		Completing the square is a method that can be used to transform a quadratic equation in standard form to vertex form.
		Any quadratic function that is not in vertex form can be put in vertex form by completing the square.
		Once in vertex form, a quadratic equation can be easily plotted by recalling graphical transformations.
	www.biology.arizona.edu /BioMath/tutorials/Quadratic/CompletingtheSquare.html (572 words)

	Extremal Values
		Let q(x) be a quadratic form in n dimensions, such that q is positive everywhere except the origin.
		The new quadratic form r(x) becomes nonpositive instead of nonnegative, but that's ok. The previous theorem still holds, and once again u vanishes on the kernel of m-e, and u is an eigen vector of m, with eigen value e.
		Using the previous theorem, one can classify a quadratic form by examining the signs of the eigen values of its associated matrix.
	www.mathreference.com /la-qf,extreme.html (650 words)

	Quadratic Forms and Inner Products (Site not responding. Last check: 2007-10-21)
		A bilinear form may be defined on a vector space when it is constructed using the VectorSpace-constructor.
		Given a matrix F belonging to Mat_K(n) that defines a bilinear form on V with respect to the current basis for V, assign F as the bilinear form associated with V. Properties of the Form
		Convert the symmetric matrix a belonging to Mat_K(n) into a polynomial, where a is interpreted as the matrix of a quadratic form.
	www.math.uiuc.edu /Software/magma/text396.html (289 words)

	All Square: Science News Online, March 11, 2006 (Site not responding. Last check: 2007-10-21)
		This result came to be known as the "15 theorem." It serves as a filter, separating a certain group of quadratic forms into universal and nonuniversal expressions.
		The universal quadratics previously recognized by Lagrange and Ramanujan belong to the matrix-defined subset considered by Conway and Schneeberger.
		Conway, J.H. Universal quadratic forms and the fifteen theorem.
	www.sciencenews.org /articles/20060311/bob9.asp (1825 words)

	Mathematical Programming Glossary (Site not responding. Last check: 2007-10-21)
		Typically, Q is presumed symmetric, in which case its gradient is 2x'Q, and its eigenvalues are real.
		Classically, this is to optimize a quadratic function over a polyhedron, defined by linear equations and/or inequalities:
		A QP is convex if its quadratic form matrix (Q) is negative semi-definite.
	glossary.computing.society.informs.org /second.php?page=Q.html (330 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		A (binary integral) quadratic form is an expression ax^2 + bxy + cy^2 where a, b and c are integers, and its discriminant is b^2 - 4ac.
		For instance 3x^2 - xy + 5y^2 is a quadratic form of discriminant -59.
		Each ewquivalence class of positive definite forms contains exactly one reduced form, so that h(n) for negative n is the number of reduced forms of discriminant n.
	www.math.niu.edu /~rusin/papers/known-math/98/classno2 (762 words)

	[No title]
		The quadratic formula is derived, in fact, from completing the square in the general case.
		The method is the same as used to solve a quadratic equation, but now is done in place, or all on the right-hand side of the function.
		And, just as the quadratic formula results from solving the general case quadratic, we can complete the square in place one last time, and derive a useful method for finding the vertex of any parabola quickly using
	www.valleyview.k12.oh.us /vvhs/dept/math/quadshelpcthes.html (529 words)

	Reduction of an indefinite binary quadratic form (Site not responding. Last check: 2007-10-21)
		, we use the PQa continued fraction algorithm to determine a reduced form and thence a cycle of reduced forms.
		99-105, Dover 1957, where the connection between a cycle of reduced forms and the periodic expansion of a reduced quadratic irrational is described.
		We generalise this, making use of the fact that the complete convergents of a quadratic irrational eventually become reduced.
	www.numbertheory.org /php/reduce.html (246 words)

	quadratic_form (Site not responding. Last check: 2007-10-21)
		quadratic_form is a class which represents binary quadratic forms, i.e., polynomials of the form f(X, Y) = a X^2 + b X Y + c Y^2 IN Z[X, Y].
		ct quadratic_form(const bigint and a, const bigint and b, bigint and c) initializes with the quadratic form (a,b,c).
		ct quadratic_form(const quadratic_ideal and A) initializes with the quadratic form associated with the ideal A.
	www.math.psu.edu /local_doc/LiDIA/node85.html (1597 words)

	Quadratic Functions in Standard Form
		This form of the quadratic function is also called the vertex form.
		C - From vertex form to general form with a, b and c.
		Rewriting the vertex form of a quadratic function into the general form is carried out by expanding the square in the vertex form and grouping like terms.
	www.analyzemath.com /quadratics/quadratics.htm (970 words)

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