A bilinear form over an $F$-vector space $V$ is a mapping $B:V\times V\to F$ that is linear in each of its arguments, when the other argument is held fixed.

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Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
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Lagrange Method for Presenting Bilinear form as sum of squares

I have the following question in my assignment which I'm having a hard time solving. For the following bilinear form, present find a digonal form (diagonal matrix form): What I thought to do at ...
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Bilinear form and cross product in hyperbolic geometry

I'm reading Patrick J. Ryan's Euclidean and non-Euclidean geometry, page 152. There is a bilinear form defined by $b\left( {x,y} \right) = {x_1}{y_1} + {x_2}{y_2} - {x_3}{y_3}$ on ${\mathbb{R}^3}$ and ...
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25 views

Bilinear Form Non-Degenrate on a Subspace

I am trying to prove the following standard result: Let $V$ be a finite dimensional vector space over a field $F$ and $f:V\times V\to F$ be a symmetric bilinear form on $V$. Let $W$ be a subspace ...
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Symmetric bilinear forms, quadratic forms and matrices

I have computed B=$ \left( \begin{array}{ccc} 0 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 1 \end{array} \right) $ Is this correct? If so, even though I may have achieved the correct ...
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1answer
26 views

Eigenvectors, bilinear forms and orthonormal bases

I have calculated (a) to be $(1,-2,2)^t, (-2,1,2)^t, (2,2,1)^t$. For (b) I have made all of these of unit length ie taken 1/3 of each vector. I have verified these are orthonormal by checking ...
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Computing the basis of a bilinear form

I am struggling to answer part (a), I don't see how I get this into matrix form? (b) is straightforward. For (c) I must use $C^t*B*C$ where B is the matrix composed of the basis vectors, so this is ...
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61 views

When are sesquilinear forms actual inner products?

I read it was enough (and necessary) to have $\overline A^T=A$ and $A$ non-singular, for a sesqui-linear form on $\mathbb{C}^n$ to be an actual inner product. (Here $A$ is a matrix for the ...
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Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
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36 views

Two versions of Lax-Milgram theorem

I'm having some troubles differentiating between two versions of Lax-Milgram theorem, one shown in my class and one that I saw is common on the internet. Let $H$ be hilbert space, $B$ bilinear form ...
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Diagonalisability of Self-Adjoint Operators for Non-Symmetric Metrics

Let $V$ be a finite dimensional vector space and $(\cdot,\cdot)$ a non-degenerate bilinear form. When $(\cdot,\cdot)$ is symmetric, every self-adjoint operator on $V$ is diagonalisable. What happens ...
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What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
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1answer
46 views

Conjecture that $A^{T}BA = ABA^{T}$ for any symmetric matrix $B$ in $\mathbb{R}^n$

While trying to understand the Kalman filter, and by experimentation with Python I came up with the conjecture in the title. First of all is it true? Second, if it is, how can I prove this? I would ...
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If I define a map on basis elements from $A\times A$ to $A$, can I extend this to a bilinear map?

This may seem a silly question but I have a map $f:A\times A\to A$ where $A$ is the dual of the augmentation ideal of $\mathbf{Z}[C_n]$ (so it has a $\mathbf{Z}$-basis). This map is defined for the ...
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In a vector space $V$, If $v^tAw = v^tBw$ for all $v,w \in V$, does it imply $A=B$?

OK. So, basically this question came up when I was trying to solve a homework question about how the matrix representation of a bi-linear form on $V$ changes if we change the basis on $V$. Let's say ...
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A dimension relation on non-degenerate bilinear forms

Let $E$ be a finite dimensional $k$ -vector space and $F$ a subspace. Let $f:E\times E \to k$ be a bilinear form which is non-degenerate on its restriction to $F$. Is it true that ${\rm dim } F + {\rm ...
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41 views

Average value of a bilinear map on a Euclidean sphere

Let $(V, g = \langle \cdot, \cdot \rangle)$ be a Euclidean vector space and $B : V \times V \to \mathbb{R}$ be a symmetric bilinear form. I would like to know if something like this is true: ...
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66 views

Explicit Isomorphism between Vector Spaces

Let $V$ and $W$ be two finite dimensional spaces. I want to show that I have a canonical isomorphism from the space of bilinear forms $\mathcal{B}= \left\lbrace B: V^* \times W^* \rightarrow ...
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1answer
20 views

Non-singular bilinear forms over a vector space.

I have a field $k$ and a finite dimensional $k$-vector space $E$. Let $f$ be a symmetric $k$- bilinear form on $E$. I define $f$ to be non-degenerate if $f(x,y)=0$ $\forall y\in E$ implies $x=0$. I ...
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Notation question about scalar products and bilinear forms

Quick notation question. Is it necessary to distinguish between a scalar product and say a bilinear form $A: V \times V^* \rightarrow \mathbb{R}^n$. Would it be recommended that say you define ...
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Conic section: What is the coordinate matrix of its bilinear form?

Given is the conic section $x^2 + xy + y^2 + 2x +3y - 3 = 0$. I need to find the coordinate matrix $M_\beta(s)$ of the bilinear form $s: \mathbb{R}^2 \times \mathbb{R}^2 -> \mathbb{R}$. I read ...
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Calculating the signature of matrix A?

given is a symmetrical bilinearform s that has the following matrix: $A = M_\beta(s) = \begin{pmatrix} -3&0&-1\\0&-3&0\\-1&0&-1\end{pmatrix}$ I have to calculate the ...
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40 views

Linear transformations in bilinear form

Be $f:V \times V \to F$ a bilinear pattern and $V$ of finite dimension. Is it correct that for every linear transformation $T:V \to V$ exists another linear transformation $T':V \to V$ for which: ...
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Understanding a bilinear form problem from Greub's Multilinear Algebra

I read the following problem from exercise sets of Greub's Multilinear Algebra, Chapter I, Sec. 1 Let $E$, $E^*$ be a pair of dual spaces and assume that $\mathit{\Phi}:E^{*}\times E\to\Gamma$ is ...
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1answer
51 views

If $A \succeq B$ is it true that $B^{-1} \succeq A^{-1}$

If $A$ and $B$ are two positive definite matrices such that $A - B$ is nonnegative definite, is it true that $B^{-1} - A^{-1}$ is positive definite? The doubt came to me when working with confidence ...
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72 views

Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
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51 views

Linear Algebra, Quadric form

Question from an exercise Let $V$ be a vector space over a field $F$ with $charF\neq2$. If $\varphi,\psi\in V^{\vee}$ are linear functionals, we will define $\varphi\cdot\psi \colon V \rightarrow ...
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219 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
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$\ker(A)=\text{Im}(A^*)^\perp$

How do I show that $\ker(A)=\text{Im}(A^*)^\perp$ for any square matrix $A$. I have done this problem before with the linear operator $T$ on a hermitian space but I can't seem to apply what I have ...
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Linear Algebra, Quadric Form, Bilinear Form

I have a question from an exercise. So $V$ is a vector space with quadric form $q:V\to \mathbb{R}$ . I have to prove that if the exists $u$, $v$ in $V$ such that $q(v)>0$ and $q(u)<0$ then ...
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If $x^T\!Ay=0$ for all $x,y $ in $ \mathbb{R}^n$ then $A=0$

Is it true that if $x^T\!Ay=0$ for all $x,y $ in $ \mathbb{R}^n$ then $A=0$. If so how do I justify this statement?
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Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
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Is there a closed form solution for slope lines of bilinear function?

Given a bilinear function $f(x,y) = a + bx + cy + dxy$, is there a closed form solution for a slope line passing through point $(x_0, y_0, f(x_0, y_0))$? It can exclude degenerate cases, e.g. $b = c = ...
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42 views

How to maximize this function

We are in an euclidian space, and we have to maximize the quadratic form : $x\in B\rightarrow (x|u) (x|v) $where $u$ and $v$ are two given vectors, and $B=\{x:||x||\leq1\}$ I don't find where i have ...
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Bilinear form over finite space

let $I$ be an ideal in local noetherian domain $(R,m)$ such that $I/I^2$ and $m/m^2$ are $R/m$ vector spaces , is there any nondegenerated symetric bilinear form on this spaces or the relation between ...
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Prove that a function $F$ is bilinear.

This time I brought a different problem. I'm starting the study of bilinear forms and I came across this question. This is probably simple, however still confuses me a bit. I must prove that ...
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How do i find a signature of a quadratic form? Also how do i represent a quadratic form as a sum/difference of squares?

For example given $(x,y,z,t) = xy+ y^2+ yz+z^2+zt$ How do i represent it as a sum and difference of squares (i.e. in the form $\sum a_iA_i^2$) and how do i find its trace? Or if i have a quadratic ...
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Non-central proper normal subgroups of unitary groups over fields

Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple? Let $F$ be a (commutative, associative, ...
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2answers
39 views

If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that

If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that $$\lim\limits_{(h,k) \to (0,0)} \dfrac{|f(h,k)|}{|(h,k)|} = 0$$.
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Question about bilinear form [duplicate]

Prove that every bilinear form $f:\mathbb R^n \times \mathbb R^n\rightarrow \mathbb R$ has a basis $\{v_1,\ldots,v_n\} \subset \mathbb R^n$ such that $f(v_i,v_j)=-f(v_j,v_i)$ for every $i\neq j$. I ...
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Exam question: need to prove that for every bilinear form there's a basis that according to it, the bilinear form matrix is anti-symmetric [duplicate]

I need to prove that for every $$f:R^{n}\times R^{n} \to R $$ bi-linear form there is a basis $\{v_1,...v_n\}$ such that if $i$ is different from $j$ then $$ f(v_{i},v_{j}) = -f(v_{j},v{i}). $$ ...
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Prove that for every $f$, a bilinear form, there exists a basis ${v_1,…,v_n}$ so that $f(v_i,v_j) = -f(v_j,v_i)$

I didn't want to bloat the title, i'll also add that the basis is in ${\mathbb{R}^n}$. This question seems kind of easy, but its from a test so I assume there is a catch here somewhere. I tried to ...
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1answer
37 views

Generalized Poincaré Inequality

In my numeric script it states for a bounded domain $\Omega \subset\mathbb{R^n}$ and any map $0 < a_0 \leq a(x) \leq a_1 < \infty $ for $x\in \Omega$ it exists a $\gamma>0$ such that the ...
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Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
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47 views

Is Cea's lemma sharp?

Given a problem in weak formulation $$ \begin{align} \text{find $u\in V$ s.th. for all $v\in V$} \\ a(u,v) = f(v) \end{align} $$ with bilinear form $a:V\times V\rightarrow\mathbb{R}$, bounded with ...
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61 views

Homework: canonical form of quadratic form

X=(x,y,z) Q(X) = $x^2 + 4xy + 6xz + 3y^2 +8yz +5z^2 $ I got by using completing the square method: Q(X) = $(x+2y+3z)^2 - (y+2z)^2$ so as I learned now I do: $u = x+2y+3z$ $v = y+2z$ $w = 0 $ ...
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Find canonical form of bi-linear form on polynomials

$V$ is vector space of polynomials in degree less or equal than $2$, we define the bi-linear form: $f(p,q) = p'(-1)q(2$) where $p$ and q are polynomials from $V$. I need to find the canonical form ...
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1answer
32 views

Scalar / Dot product

I have a simple question about Scalar / Dot product. (http://en.wikipedia.org/wiki/Dot_product) Say f is a bilinear form. I have to tell if f defines a dot product. I didn't understand what I should ...
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63 views

Prove that $A^{t}A$ is positive definite

$A$ is an invertible matrix over $\mathbb{R}$ (nxn). Show that $A^{T}A$ is positive definite. I looked up for it and found this two relevent posts but still need help. positive definite and transpose ...
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1answer
81 views

Structure of $G$-invariant bilinear forms over (finite?) fields

I have a question about the structure of $G$-invariant bilinear forms. Let $G$ be an arbitrary finite group and $\mathbb{F}_q$ a finite field such that $2|G|$ is not divisible by the characteristic of ...