What is Quadratic forms: Definition and 41 Discussions

In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,








{\displaystyle 4x^{2}+2xy-3y^{2}}
is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If



{\displaystyle K=\mathbb {R} }
, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form.
Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form).
Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials.

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  1. C

    I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

    I learned that for a bilinear form/square form the following theorem holds: matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form. Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...
  2. A

    I Canonical Form for quadratic equations *with* linear terms

    Hello: I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables: $$ax^2+by^2+cxy+dx+ey+f=0$$ Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part): $$ w^TDw+[d \ \ e]w+f=0$$ $$w^TDw+Lw+f=0$$ where $$...
  3. S

    I Find the minimum and maximum value of a quadratic form

    By working with the following definition of minimum of a quadratic form ##r(\textbf{x})##, ##\lambda_1=\underset{||\textbf{x}||=1}{\text{min}} \ r(\textbf{x})## where ##\lambda_1## denotes the smallest eigenvalue of ##r##, how would one tackle the above problem? It is clear that the diagonal...
  4. G

    Reduced equation of quadratic forms

    Homework Statement Given the following quadric surfaces: 1. Classify the quadric surface. 2. Find its reduced equation. 3. Find the equation of the axes on which it takes its reduced form. Homework Equations The quadric surfaces are: (1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0 ##...
  5. G

    I Can Quadratic Forms Map Integers to Integers?

    Alright, so this might be a stupid question, but nevertheless, I ask. I am to consider whether the quadratic form ## P(x,y) = a x + b y + d xy ## can map the integers onto the integers. So through a change of basis, I re-express this as ## P'(u,v) = Au^2 + Bv^2 ## for rational A and B...
  6. Euler2718

    I Clarifying a corollary about Quadratic Forms

    The question comes out of a corollary of this theorem: Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...
  7. L

    Identify the quadratic form of the given equation

    <Moderator's note: Moved from a technical forum and thus no template.> Hello I am given the following problem to solve. Identify the quadratic form given by ##-5x^2 + y^2 - z^2 + 4xy + 6xz = 5##. Finally, plot it. I cannot seem to understand what I have to do. The textbook chapter on...
  8. A

    Expressing a quadratic form in canonical form using Lagrange

    Problem: Express the quadratic form: q=x1x2+x1x3+x2x3 in canonical form using Lagrange's Method/Algorithm Attempt: Not really applicable in this case due to the nature of my question The answer is as follows: Using the change of variables: x1=y1+y2 x2=y1-y2 x3=y3 Indeed you get...
  9. mr.tea

    Quadratic forms under constraints

    Homework Statement Find the minimum value of ## x_1^2+x_2^2+x_3^2## subject to the constraint: ## q(x_1,x_2,x_3)=7x_1^2+3x_2^2+7x_3^2+2x_1x_2+4x_2x_3=1 ## Homework EquationsThe Attempt at a Solution I am not really sure how to think about it. I have seen the opposite way but have not seen this...
  10. E

    Quadratic forms and kinetic energy

    I heard that proportionality of kinetic energy with square of velocity, ##E_k\propto v^2##, can be derived with help of quadratic forms. It goes like: we guess that ##E_k\propto v^2## and we assume that momentum ##p\propto v##, then equation is valid in another inertial system. And so on. The...
  11. E

    Quadratic form is nover zero?

    Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true. True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.
  12. E

    MHB Diagonalizing quadratic forms in WolframAlpha

    Hello, Suppose I have a vector space $V$ over $\Bbb R$, a quadratic form $f(x)$ over $V$, some basis of $V$ and a symmetric matrix $A$ corresponding to $f$ in that basis, i.e., $f(x)=x^TAx$. Using, for example, the Lagrange method, I can find a change-of-basis matrix $C$ ($x=Cx'$) such that in...
  13. F

    MHB Classify Quadratic Surfaces: Ellipsoids, Hyperboloids, Paraboloids & Cylinders

    On the basis of the eigenvalues of A, classify the quadratic surfaces X'AX+BX+k=0 into ellipsoids, hyperboloids, paraboloids and cylindres. Can somebody help me to solve the problem?
  14. caffeinemachine

    MHB A theorem on Quadratic Forms in Reid's Book not at all clear.

    Hello MHB, I have been reading a book on Algebraic Geometry by Reid. On page 15, there's a theorem on Quadratic forms. The book doesn't explicitly define what a Quadratic Form is. From Hoffman & Kunze's book on Linear Algebra I found that given an inner product space $V$ over a field $F$, the...
  15. matqkks

    MHB Quadratic Forms: Beyond Sketching Conics

    What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
  16. matqkks

    Quadratic Forms: Beyond Sketching Conics

    What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
  17. marellasunny

    Where would I use quadratic forms and how?

    Wiki defines :In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. Yes,all nice and dandy,I get to then express it in terms of matrices and then I find the eigen values and then find the canonical quadratic form,the usual boring linear algebra...
  18. E

    Linear algebra question, quadratic forms.

    A is a square matrix. x, b are vectors. I know for Ax=b, that given b, there are an infinite number of pairs (A, x) which satisfy the equation. I'm wondering if the same is true for xAx=b. in particular, what if (x, A, b) are all stochastic vectors/matrices (i.e the entries of b and x add to...
  19. ElijahRockers

    Quadratic Forms & Lagrange Multipliers

    Homework Statement I'm having trouble grasping http://www.math.tamu.edu/~vargo/courses/251/HW6.pdf. Our teacher has decided to combine elements from Linear Algebra, and understanding Quadratic forms with our section on lagrange multipliers. I am barely able to follow his lectures. If I look...
  20. T

    Quadratic forms and sylvester's law of inertia

    Say I start with a quadratic form: x^2 - y^2 - 2z^2 + 2xz - 4yz. I complete the square to get: (x+z)^2 - (y+2z)^2 + z^2. (So the rank=3, signature=1) The symmetric matrix representing the quadratic form wrt the standard basis for \mathbb{R}^3 is A =\begin{bmatrix} 1 & 0 & 1 \\...
  21. O

    Simplifying quadratic forms

    I'm having a bit of a brain fart here. Given a positive definite quadratic form \sum \alpha_{i,j} x_i x_j is it possible to re-write this as \sum k_i x_i^2 + \left( \sum \beta_i x_i \right)^2 with all the ki positive? I feel like the answer should be obvious
  22. B

    Classifying Symmetric Quadratic Forms

    Hi, All: I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space over the reals. My idea is to use the standard basis for R^3 , then use the matrix representation M =x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M...
  23. W

    Linear Algebra and Quadratic Forms

    Homework Statement For the quadratic form x2-2xy+2yz+z2: a) Find a symmetric matrix that allows the quadratic form to be written as xTAx. b) Determine if the critical point at the origin is a minimum, maximum, or neither. c) Find the points for which the quadratic form achieves its...
  24. L

    2 dimensional quadratic forms

    I am told that the set of positive definite quadratic forms on R^2 has a metric that turns it into H x R where H is the hyperbolic plane. Can you describe this metric? * As a space the forms are viewed as GL(2,R)/O(2).
  25. B

    Quadratic Forms: Closed Form from Values on Basis?

    Hi, Everyone: I have a quadratic form q, defined on Z<sup>4</sup> , and I know the value of q on each of the four basis vectors ( I know q is not linear, and there is a sort of "correction" for non-bilinearity between basis elements , whose values --on all pairs (a,b) of...
  26. M

    Summing Quadratic Forms in Three Variables: True or False?

    Homework Statement True or False and Why? "The sum of two quadratic forms in three variables must be a quadratic form as well." Homework Equations q(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2+x_1x_3+x_2x_3 The Attempt at a Solution I am definitely missing something. To me this is a...
  27. R

    Quadratic Forms for SL(2;R)

    Homework Statement Construct the analytic mapping \phi(x,y) for the H^{2+} \times S^1 representation of SL(2;R) Homework Equations g(x) \circ g(y) = g(\phi(x,y)) The Attempt at a Solution So, all points in SL(2;R) lie on the manifold H^{2+} \times S^1. I also know that SL(2;R) is...
  28. I

    Quadratic forms and congruence

    Homework Statement How many equivalence classes under congruence (as in two quadratic forms - n dimensional vector space over field - being congruent if one can be obtained from the other by a change of coordinates) are there when (i) n=4 and field is complex numbers (ii) n=3 and field is real...
  29. A

    Quadratic forms with p-adic coefficient

    How can I found out in which p-adic fields a quadratic form represent 0? For example in which p-adic fields does the form 3x2+7y2-15z2 represent zero?
  30. C

    Linear Algebra, Quadratic Forms, Change of Variable (concept)

    Homework Statement Make a change of variable that transforms the quadratic form with no cross-product term: 9x1^2 - 8x1x2 = 3x2^2 Homework Equations A = PDP^-1 Q = y^TDy The Attempt at a Solution I know the answer. This is a question regarding concept. The eigenvalues for...
  31. K

    Number theory: Binary Quadratic Forms

    P.S. I'm not sure where to post this question, in particular I can't find a number theory forum on the coursework section for textbook problems. Please move this thread to the appropriate forum if this is not where it should belong to. Thanks!
  32. W

    Positive definite quadratic forms proof

    Homework Statement Given a real symmetric matrix A, prove that: a) A is positive definite if and only if A = (B^T)B for some real invertible matrix B b) A is positive semidefinite if and only if there exists a (possibly singular) real matrix Q such that A = (Q^T)Q Homework Equations...
  33. S

    Quadratic Form Q: Matrix A & Lambda Calculation

    Let Q: R3 \rightarrow R be the quadratic form given by Q(x) = 2x1x2 + 2x1x3 + 2x2x3 where x = (x1x2x3)t How do I write down the matrix A of the quadratic form Q in the standard matrix E. and how do I find the numeric values for \lambda
  34. C

    Diagonal Quadratic Forms of a Matrix

    Homework Statement Let the quadratic form F(x,y,z) be given as F(x,y,z) = 2x^2 + 3y^2 + 5z^2 - xy -xz - yz. Find the transitional matrix that would transform this form to a diagonal form. Homework Equations A quadratic form is a second degree polynomial equation in three...
  35. S

    Quadratic forms, diagonalization

    Can a quadratic form always be diagonalised by a rotation?? Thx in advance
  36. A

    Counting Quadratic Forms on Fp^n: Exploring the Field of p Elements

    For an odd prime number p let Fp be the field with p elements, ie. the integers {0...,p-1} with addition and multiplication defined modulo p. How many quadratic forms are there on the vector space Fp^n I don even know how to start this question
  37. M

    Quadratic forms of symmetric matrices

    hi i just wanted a quick explanation of what a symmetric matrix is and what they mean by the quadratic form by the standard basis? (1) for example why is this a symmetric matrix [1 3] [3 2] and what is the quadratic form of the matrix by the standard basis? (2) also how would i go...
  38. K

    Positive definite real quadratic forms

    Q: Suppose q(X)=(X^T)AX where A is symmetric. Prove that if all eigenvalues of A are positive, then q is positive definite (i.e. q(X)>0 for all X not =0). Proof: Since A is symmetric, by principal axis theorem, there exists an orthogonal matrix P such that (P^T)AP=diag{c1,c2,...,cn} is...
  39. T

    Conditional Probabilities relating quadratic forms of random variables

    Well I'm getting pretty frustrated by this problem which arose in my research, so I'm hoping someone here might set me on the right track. I start with n random variables x_i, i=1..n each independently normally distributed with mean of 0 and variance 1. I now have two different functions...
  40. B

    Proof of Binary Quadratic Forms and Modulo Squares with Hint

    show that if a number n is represented by a quadratic form f of discriminant d then 4an is a square mod |d|. I have no clue how to even start this proof. I tried using the jacobi symbol.. but it's not gettin me anywhere. Could someone give me a hint.. :confused:
  41. M

    Quadratic forms, linear algebra

    I have a question that i have to do, the only problem is time. Since i have to finish my stats assignment, would anyone tell me the steps involved in solving this problem(in order), Rotate and translate the coordinate axes, as necessary to bring the conic section 3x^2 -8xy -12y^2 -30x-64y=0...