Questions tagged [bilinear-form]

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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Lattice corresponding to the incidence matrix of a weighted tree is indecomposable

Suppose $\Gamma$ is a (necessarily connected) weighted tree with $k$ vertices $v_1,\dots,v_k$. Suppose the weight of $v_i$ is $n_i\in \Bbb Z$. Let $A$ be the $k\times k$ symmetric matrix defined by $...
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Positiveness of Bilinear form

I would like to show that $$A = x^T G(x) \tanh(x) \geq 0$$ (with equality only for $x=0$), for $x\in\mathbb{R}^n$. It is known that $G(x)\in\mathbb{R}^{n\times n}$ is a symmetric and positive definite ...
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Find an orthogonal basis for vector space of $2\times2$ real valued matrices

Let $V$ be the vector space of $2\times2$ real valued matrices. And $(A, B) \to \operatorname{tr}(A^\top \cdot B^\top)$ Find an orthogonal basis for $V$. I can't figure out how to make $$ \begin{...
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When is the restriction of this bilinear form non-degenerate?

In Jürgen Elstrodt's Measure and Integration Theory (8th German edition) there is the following exercise I.3.6: Exercise: The group $(\mathfrak{P}(X), \Delta)$ (power set with symmetric difference) is ...
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Find a basis for which the matrix of the following bilinear form has the form described in Sylvester's theorem

Let V be the vector space of real valued polynomials with degree smaller equal 3 and let $$(p,q) = \int_{-1}^{1} p(x)q(x) \,dx - 100p(0)q(0)$$ Is the bilinear form degenerate and find a basis for V in ...
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Prove this is a bilinear form

I'm dealing with an exercise on bilinear forms, but I don't know how to properly start. I have to prove that $\beta(A, B) = \mathrm{tr}(AB) - \mathrm{tr}(A)\mathrm{tr}(B)$ is a bilinear form, where $A,...
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Identifying linear operator with a bilinear symmetric form using Theorem of Schwarz

We study the energy functional $E$ of the form $$E(v)=\frac{1}{2}a(v,v)+\int_{\Omega}F(x,v).$$ Let $V$ be a real Banach space with norm $||\cdot||_{V}$ and denote by $V^{'}$ the dual space. By $\...
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Totally isotropic subspace for bilinear pairing over ring

Consider the following well-known inequality: Let $b$ be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb{F}$-vector space $V$ and $W$ a totally isotropic subspace. ...
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Can every Bilinear map $B:V\times V \to k$ be written as follows?

I am studying representation theory on my own and encountered tensor product spaces. I learnt that the space of Bilinear maps on $V\times V$, for a vector space $V$ is isomorphic to $V^* \otimes V^*$ ...
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If $f:\,V\longrightarrow W$ preserves bilinear form then $f$ is linear.

Let $V$ and $W$ be $n$ dimensional vector spaces on $\mathbb R$ with bilinear forms $\langle,\rangle_V:\, V\times V\longrightarrow\mathbb R$ and $\langle,\rangle_W:\, W\times W\longrightarrow\mathbb R$...
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Uniqueness of codimension one embedding of a lattice of square-free determinant into the standard lattice

This question is a continuation of my previous question: Uniqueness of codimension one embedding into the standard lattice. Let $L$ be a positive-definite lattice of rank $n$, and suppose $L$ embeds ...
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Matrix-free definition for an operation on symmetric positive semidefinite bilinear forms

Let $SPD_2(\mathbb{R}^n)$ be the set of all symmetric and positive semidefinite bilinear forms $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$. Define a binary operation $(f,g) \mapsto f \ast g$ ...
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Alternative description of generalized orthogonal group.

I am studying Lie groups and Lie algebras.One of the standard examples of Lie groups is $O(n;k)$,which is called the generalized orthogonal group.It is defined by $O(n;k)=\{T\in GL_n(\mathbb C): [Tx,...
Kishalay Sarkar's user avatar
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Maximal totally isotropic subspace in a space with a degenerate skew-symmetric bilinear form

Let $V$ be a real finite-dimensional vector space with a skew-symmetric bilinear form $B \colon V \times V \to \mathbb{R}$. In general, we assume that the form $B$ is degenerate. A subspace $S \subset ...
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Show that the trace bilinear form is nondegenerate

Consider a simple Lie algebra $\mathfrak{g}$ which contains a subalgebra isomorphic to $\mathfrak{sl}(2,\mathbb{R})$ and its nontrivial irreducible representation $\pi:\mathfrak{g}\to\mathrm{End}(V)$. ...
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Where did we used the condition $\langle , \rangle: A \times A \rightarrow \mathbb{Q}/\mathbb{Z},$ is alternative to prove $A$ has square cardinatiry?

Lemma 4. Let $A$ be a finite abelian group. If there is a non-degenerate alternating bilinear pairing $$\langle , \rangle: A \times A \rightarrow \mathbb{Q}/\mathbb{Z},$$ then, $A \cong S \times \hat{...
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Let $S$ be the subgroup of $A$ such that $\psi(s,s')=0$ for all $s,s'\in S$ and $S$ is maximal with respect to this property.

Let $A$ be a finite abelian group, and let $$ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $$ be an alternating, non-degenerate bilinear form on $A$. Let $S$ be the subgroup of $A$ such that $\psi(s,s')...
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Uniqueness of codimension one embedding into the standard lattice

Let $X$ be a positive definite lattice of rank $n$, and suppose $X$ embeds into the standard lattice $\Bbb Z^{n+1}$. Then is it true that the embedding is determined uniquely up to automorphism of $\...
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Bilinear Form inducing Canonical Isomorphisms

Let's say $V$ and $W$ are two finite dimensional vector spaces over $\mathbb R$. A bilinear form $\mathrm B : V \times W \to \mathbb R$ is necessarily linear in both factors: $\mathrm B(v_1+v_2,w) = \...
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Tranpose of a mixed tensor

If $(-)'$ denotes the transpose operation then for a type (1,1) mixed tensor ${T^a}_b$, $\left({T^a}_b\right)'={T^b}_a$. Although this seems right I would like to be able to show this starting with a ...
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Brezis' exercise 8.25.2: prove that the bilinear form $a$ on $H^1 (I)$ is coercive

Let $I$ be the open interval $(0, 2)$ and $V=H^1(I)$. Consider the bilinear form $$ a(u, v) = \int_0^2 u'v' + \left(\int_0^1 u\right) \left(\int_0^1 v\right) . $$ I'm trying to solve a problem in ...
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Projections in semi-normed spaces

I have a question regarding the projection onto a finite-dimensional subspace of a semi-normed vector space: Let $V$ be a real vector space (either finite or infinite-dimensional) and let $\langle\...
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Show there exists $T$* such that $F(T(u), v) = F(u, T^{*}(v))$

I'm having trouble with the following problem: Let V be a vector space of finite dimension and F a non-degenerate symmetric bilinear form on V. Show that if $T \in L(V,V)$, then there exists $T$* $\in ...
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Weird quadratic form exercise

Let $H$ be the quadratic form in vector space $\mathbb{R}^3$ s.t $$H=x^2+2y^2+3z^2+2xy+2xz+2yz.$$ Let $\varphi$ be the symmetric bilinear form on $\mathbb{R}^3$ and $\varphi:\mathbb{R}^3\times \mathbb{...
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self-orthogonal binary codes

From Robert Griess's article "Elementary abelian p-subgroups of algebraic groups": (2.7) Definition. Let char$(\mathbb{K})\neq 2$ and let $V$ be an $m$-dimensional vector space with ...
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Symmetric bilinear form from Hessian

Assuming a PDE is given by the Hesse-operator $$ -\nabla \otimes \nabla \lambda = \mathbf{F} \, , $$ then testing the left-hand-side with a symmetric tensor $\mathbf{S}$ leads to $$ -\langle \mathbf{...
Adam's user avatar
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Compare eigenvalues of comparable matrices

Consider two symmetric matrices $A,B$ on $\mathbb{R}^n$ and assume that $A\geq B$ as bilinear forms, i.e. $A(x,x)\geq B(x,x)$ for any vector $x$. Let $(\lambda_i)$ and $(\mu_i)$ be the ordered (...
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Find the max and min of $(a \mathbf{x} + c)(b \mathbf{x} + d)$, where $\mathbf{x}$ is a vector with linear constraints.

Suppose there is a $n$-dimensional column vector $\mathbf{x}$, and an objective function $f(\mathbf{x}) = (a \mathbf{x} + c)(b \mathbf{x} + d)$, where $a$ and $b$ are $n$-dimensional row vector; $c$ ...
Augustin Pan's user avatar
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A Question About the Proof on Existence of Orthonormal Basis of a Nondegenerate Bilinear Form

Link For non-degenerate symmetric bilinear form there is a basis $\{v'_i\}$ such that $B(v_i, v'_j) = \delta_{ij}$ In the thread above, the proof is based on induction on the dimension of the ...
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Optimization problem: x and y that minimizes x^H A y

I have a problem formulated as: $\underset{x,y}{\min} x^HAy$ subject to $\sum x = 1$ and $\sum y = 1$. Where A is a complex matrix, and x and y are column vectors that I need to find. This is very ...
Håvard Arnestad's user avatar
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Dimension of an isotropic subspace

I am trying to prove the following statement about isotropic subspaces. The definition I have is: Let $V$ be a finite-dimensional vector space on which there is defined a non-degenerate bilinear $B$. $...
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$f \mapsto b, b(r,m)=r\cdot f(m)$ gives an isomorphism of $R$-modules $\Phi_N : \text {Hom}_R(M,N) \rightarrow \text {Bilin}_R(R×M,N)$

Let $R$ be a commutative ring $M$ and $N$: $R$-modules Let $f : M \rightarrow N$ is a homomorphism, I have shown that the map $b: R\times M \rightarrow N$ given by $b(r,m) = r \cdot f(m)$ is R-...
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Double Orthogonal Complements in a Bilinear Form with a Specific Matrix

I'm studying a problem from linear algebra: For a bilinear form $b: V \times V$ and a subvector space $U $ in $ V $ we set $ U^{\perp}:=\{v \in V \mid b(u, v)=0 \text { for all } u \in U\} . $ Now let ...
Marius Lutter's user avatar
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Fraction of bilinear forms vs bilinear form of fraction and the formulation of the objective function of Linear Discriminant Analysis

Let $A, B \in M(n, \mathbb{R})$ be $n \times n$ symmetric square matrices with real values and suppose $B$ is invertible. We can define three bilinear forms $(u,v) \mapsto u^TAv$ $(u,v) \mapsto u^TBv$...
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Closed-form solution to maximization of rational function with linear and bilinear form [closed]

I have the function: $$f(\mathbf{x}) = \frac{\mathbf{x}^\mathrm{H}\mathbf{w} + \mathbf{w}^\mathrm{H}\mathbf{x}}{c + \mathbf{x}^\mathrm{H}\mathbf{A}\mathbf{x}},$$ where $\mathbf{x},\mathbf{w}\in\mathbb{...
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Definitions of non degenerate bilinear forms

Referring to this Wikipedia page, The definition of a non degenerate bilinear form is given as a bilinear form $f(x,y)$ such that $v \to (x \to f(x,v))$ is an isomorphism. We are also told that the ...
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Bilinear form associated to coxeter system

Let $(W, S) $ be a coxeter system where $S=\{s_1,…, s_n \} $ and the order of $s_is_j$ is denoted $m_{ij}$. Let $V $ be a real vector space with basis $u_1,…, u_n $. Let $B$ be the bilinear form ...
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Show that the matrix $A$ is positive definite

I'm trying to solve the following problem: Let $A = \begin{bmatrix} 1 & 2 & 3 & -2\\ 2 & 5 & 4 & -5\\ 3 & 4 & 14 & -2 \\ -2 & -5 & -2 & ...
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Prove that the function $f(x_1 , x_2) = x_1 x_2$ is quasi-concave on $S = \mathbb{R}_{++}^2$

Prove that the function $f(x_1 , x_2) = x_1 x_2$ is quasi-concave on $S = \mathbb{R}_{++}^2$ I started with the definition of quasi-concavity: $$ f( \lambda x + (1-\lambda) y) \geq \min \left( f(...
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$\varphi(x)= \min\{\||x-u\||^2 ; u \in U \}$ is a quadratic form

Let $V$ be a finitely generated Euclidean vector space, $U \subset V$ a $\mathbb {R}$ -subvector space. The mapping $\varphi: V \rightarrow \mathbb{R}$ is through $\varphi(x)= \min\{\||x-u\||^2 ; u \...
Marius Lutter's user avatar
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2 answers
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Adjoint map and $V \cong V^{**}$

I am confused after following these lines Note that the adjoint of a bilinear form $B:V\rightarrow V^{*}$ would have the same type $B^{*}:V\rightarrow V^{*}$ (adjoint map) (using $V^{**} \cong V$ ). ...
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Numpy/PyTorch break problem symmetry due to numeric precision

In my problem, I apply a linear transformation to a Gaussian random variable $Y \in \mathbb{R}^n$, by multiplying it with a matrix $\mathbf{f} \in \mathbb{R}^{n \times d}$, with $d<<n$. With $\...
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Check if the bilinear form given in the standard basis gives some dot product to $\mathbb R^3$

Check if the bilinear form given in the standard basis by $$\begin{bmatrix} 1 &4 &0\\ 4 &4 &0\\ 0 &0 &2 \end{bmatrix}$$ gives some dot product to $\mathbb R^3$. ...
timofiej8384's user avatar
1 vote
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Let $f:V \times W \to \mathbb F$ and $s:V \to W^*, r:W \to V^*$. Find $[s]^B_{C^*}, [r]^C_{B^*}$.

Let $f:V \times W \to \mathbb F$ bilinear map and $s:V \to W^*, r:W \to V^*$ linear transformations s.t: $s(v)(w) = f(v,w)$ and $r(w)(v) = f(v,w), \forall v \in V$ and $\forall w \in W$. Let $B$ an ...
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Showing a bilinear form is coercive and continuous

Let $\Omega \subset \mathbb{R}^3$ be open and bounded. Define $H$ to be the completion of the space divergence-free smooth functions with compact support in $\Omega$ in the $(L^2(\Omega))^3$ norm. ...
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Reflexivity of infinite dimensional vector spaces and existence of non degenerate bilinear forms

How can an infinite dimensional vector space like (most of) the Lebesgue or Sobolev spaces even be reflexive? If an infinite dimensional vector space cannot be isomorphic to its dual space, the how ...
whatever's user avatar
2 votes
1 answer
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How to extend an orthogonal linearly independent set to an orthogonal basis, with respect to a symmetric bilinear form?

Let $F$ be a field whose characteristic is not $2$. Let $X$ be an $n(<\infty)$-dimensional vector space over $F$, equipped with a symmetric bilinear form $\langle\cdot,\cdot\rangle$ on $X$. Let $...
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on a complex vector space: $\|x\| = \|y\| \Leftrightarrow x+y$ orthogonal to $x-y$

I want to know if $\|x\| = \|y\|$ $\Leftrightarrow$ $x+y$ orthogonal to $x-y$. I was able to prove it for $V$ being a real vector space. However, I'm not able to work it out for complex vector spaces. ...
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A global view-point on pseudo-Riemannian manifolds [closed]

A pseudo-Riemannian manifold $(M,g)$ is a generalisation of the concept of a Riemannian manifold where we relax positive-definiteness to non-degeneracy. $\alpha$) Non-degeneracy is still enough to ...
Fantas Anadolou's user avatar
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Non-degenerate bilinear forms and orthogonal bases for free modules

Given a finite-dimensional free module $M$ over a (commutative) ring $R$, let $(-,-)$ be a symmetric non-degenerate $R$-bilinear map $M \times M \to R$. Will $M$ admit two $R$-module bases, $e_i$ and $...
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