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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Question about $H^1$ norm

The $H^1$ norm is given by: $$(u,v)_{H^1}=\int_0^1u′v′+\int_0^1uv$$ But in a problem that I'm solving, I have a constant in the second integral: $$|a(u,v)|\le|k|||u'||_{L^2}||v||_{L^2}+\Bigg|\int_0^1u'...
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Is a scalar product positive definite on a unique maximal subspace?

Let $V$ be an $n$-dimensional real vector space and \begin{equation} \eta\colon V\times V\to\mathbb R \end{equation} a nondegenerate symmetric bilinear form. Sylvester’s Law of Inertia allows us to ...
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Show that is continuous and coercive

I have the following boundary value problem, where $k, q \in \mathbb{R}$ are given. $$-u'' + ku' +qu = f,\qquad u(0)=u(1)= 0$$ What I want to prove is that the bilinear form, associated with this BVP,...
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Calculating the signature of two hermitian forms [closed]

I have a simple question regarding the calculation of the signature of two hermitian forms. $L:\mathbb{C^4}\times\mathbb{C^4}\rightarrow \mathbb{C}$ with $L\bigg( \begin {pmatrix} x_1\\y_1\\z_1\\t_1\...
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Problem on finding Intersection form of compact,orientable $4-$manifolds .

$\mathbf {The \ Problem \ is}:$ Let $M$ be an $\mathbb{F}$-oriented manifold of dimension $2 n$ for a field $\mathbb{F}$. Consider the non-singular bilinear form $H^{n}(M ; \mathbb{F}) \otimes H^{n}(M ...
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A direct sum of Symmetric and Alternating Bilinearforms

Show that the vector space $\text{Bil}(V)$ of all bilinear forms on $V$ can be decomposed in to the direct sum of $\text{Bil}(V)_{\text{sym}} \bigoplus \text{Bil}(V)_{\text{alt}}$, where $\text{Bil}(V)...
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Image of a bilinear form

Say that a commutative ring with one $A$ has Property (P) if the image $\beta(M\times N)\subset A$ of any bilinear form $\beta:M\times N\to A$ is an ideal of $A$. Here $M$ and $N$ are two $A$-modules. ...
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Find the signature of a bilinear form given by a matrix

I'm trying to complete the bilinear form given by the matrix $$M=\left(\begin{array}{ccc}1 & -1 & 2 \\ -1 & 3 & 1 \\ 2 & 1 & 1\end{array}\right)$$ into squares to find the ...
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Non-singular Conics are equivalent to some projective line.

The proof begins on pg.22. I am having trouble understanding why ${B|}_{W_y} \not\equiv 0$. Since I couldn't understand this part of the proof (on pg. 24 of the slides), I was trying to show it ...
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Pairing on Differentials

Let $X$ be a (smooth, projective) curve of genus $g$ defined over a number field $K$, and write $\Omega^{1}$ for the $K$-vector space of holomorphic differentials. The motivation for my question comes ...
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Restriction of a quadratic form with signature $(1, n-1)$.

I want to prove the following fact. Let $f$ be a nondegenerate symmetric bilinear function on a vector space $V$ with positive index of inertia equal to 1. And let $f(x,x) > 0$ for some vector $x \...
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Definiteness of Matrix over Torus

I am interested in characterizing the definiteness of the following bilinear form over the torus: $$\langle x, A x \rangle$$ $$A = A^T, A \in \mathbb{R}^{2n \times 2n}$$ Where the inner product ...
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Determine whether smooth symmetric bilinear form is definite over the trivial normal bundle of the oblique manifold

Given a smooth symmetric bilinear form over the trivial normal bundle of the oblique manifold $\mathcal{OB}(n,m)$ imbedded in $\mathbb{R}^{n \times m}$ is there an efficient way to determine whether ...
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How to define this function in terms of tensor products

This problem was left as an exercise in algebra class of mine but I am not able to make any progress. I have been following book by atiyah and macdonald and I must say I had difficulty in ...
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Linear algebra over finite fields

Let $\mathbb{F}_p^3$ be a $3$-dimensional vector space over $\mathbb{F}_p$ with $p$ odd. For any $\mathbf{x}\in \mathbb{F}_p^3$ define its "norm" $\lVert \mathbf{x}\rVert=x_1^2+x_2^2+x_3^2,$ ...
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Checking my understanding of Einstein summation convention

I have a limited understanding of the convention so I'd like to check. Given a (0,2) tensor or bilinear map $(e_{i}\otimes e_{j})=\zeta_{ij}$ with $i,j=1,2,...,n$, and two covectors $\Omega^{1}=\sum^{...
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Is there a non-degenerate bilinear form on $R^N$ with a one dimensional kernel?

Let $M: \mathbb{R}^n\times \mathbb{R}^n \rightarrow \mathbb{R}$, $n >= 3$ be a non-degenerate bilinear form. Are there $M$, s.t. $dim(ker(M)) = 1$? What can we say in general about the dimension of ...
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Double orthogonal complement of a finite dimensional subspace w.r.t. a nondegenerate symmetric bilinear form on an infinite dimensional space

Let $U$ be a finite-dimensional subspace of an infinite-dimensional space $V$ equipped with a nondegenerate symmetric bilinear form $\phi$. Is it necessarily true that ${(U^{\perp})}^{\perp}=U$? If ...
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Show that there exists a unique solution

Help me to understanding the Lax-Milgram Theorem. I've shown that there exits a bilinear form $$ a : H_0^1([0,1]) \times H_0^1([0,1]) \to \mathbb{R}, (u,v) \mapsto \int_0^1 ( u'v' - K uv), \quad \text{...
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How to prove that $B$ is non-degenerate? does the question has typo?

Here is the question I want to prove: Let $V = M_2(\mathbb{R})$ be the space of all $2\times 2$ matrices over $\mathbb{R}.$ Show that $$B(X,Y) = det(X+Y) - det(X) - det(Y)$$ Where $X,Y \in V,$ is a ...
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Artin's theorem on Galois extension and related question on bilinear maps

Theorem: (Lang's Algebra, p. 264) Let $E$ be any field, $G$ be a finite group of automorphisms of $E$, and $F:=E^G$ be fixed field of $G$: $$ E^G=\{\alpha\in E : \sigma(\alpha)=\alpha \forall \sigma\...
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Continuity of a bilinear form on $H^1(0,2)$

Problem Statement: Let $H := H^1(0,2) := W^{1,2}(0,2)$ and define the function $B : H^2 \to \mathbb{R}$ by $$B(f,g) := \int_0^2\!\! f'g' + \int_0^1\!\! f \int_0^1\!\! g$$ Then $B$ is a bilinear form; ...
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If $f$ and $g$ two non-degenerate quadratic forms If $\delta(f)=\delta(g)$ is square in $\mathbb{F}_q$ then $f$ and $g$ are equivalent.

Let $f$ and $g$ two non-degenerate quadratic forms of a vecor space $E$ over a finite field $\mathbb{F}_q$ we note by $\delta(q)$ the discriminant of a quadratic form $q$. I want to show that if $\...
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A question about bilinear forms

Let $\phi$ be a bilinear form on a vector space $V$. Let $U$ be a vector subspace of $V$. Suppose that $\phi$ and $\phi_{|U\times U}$ are both non-degenerate. Prove that if $U$ is finite-dimensional, ...
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convert biconvex functions to convex functions

Bilinear functions on a cartesian product of vector spaces can be converted to linear functions on the tensor product of the two spaces. In the same spirit, is there a way to convert biconvex ...
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About another proof of Witt's theorem

This is a proof of Witt's cancellation theorem from Uzi Vishne's book (I wrote it in my words (in english) so if there is anything that does not seem accurate please tell me). Witt's cancellation ...
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Let V a real vector space of finite dimension, with a positive definite inner product and let A a symmetric operator in this space

I want to know how can I generalize this excersice. Let V a real vector space of finite dimension, with a positive definite inner product and let A a symmetric operator in this space a)We have $g(v,w)...
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Generalization of the Riesz representation theorem

Any finite dimensional vector space V endowed with nondegenerate bilinear form can be canonically identified with its dual space. I wonder if I can find similar identification for infinite dimensional ...
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Positive definite Hermitian form with same imaginary part

Consider a complex vector space V, if I decompose a Postive definite Hermitian form (,) into real and imaginary part, i get $$(v,w)=\lbrace v,w \rbrace +[v,w]i$$ How can I show that whenever two such ...
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Why supposing that $\operatorname{char}(F) \neq 2$?

I have a question regarding to part$(b)$ in the problem below: " Let $V$ be a finite-dimensional vector space over the field $F,$ and let $B$ be a bilinear form on V. $(b)$ Suppose $\operatorname{...
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Prove that if $B$ is symmetric then its matrix is symmetric.

Let $V$ be a finite-dimensional vector space over the field $F,$ and let $B$ be a bilinear form on V. $(a)$ $B$ is said to be symmetric if $B(v_1, v_2) = B(v_2, v_1)$ for all $v_1, v_2 \in V.$ Prove ...
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Proof about Garding's Inequality for a bilinear form

Considering the following bilinear form : $$a(\phi,\psi)=\intop_{\varOmega}\left(\sum_{g=1}^{G}D_{g}\nabla\phi_{g}\cdot\nabla\psi_{g}+\sum_{g,h=1}^{G}\varSigma_{gh}\phi_{g}\psi_{h}\right)\,\mathord{d}\...
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When will a complex representation of a semi-simple lie group, have an invariant hermitian form?

I know for compact groups, there is always one, and it can be made positive/negative definite. Does this remain true for non-compact groups, allowing for indefinite signature? I cannot think of an ...
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All invariant forms of (a representation of) a semi-simple lie group

Take for example, $SU(2)$. There are two well known invariant forms in the fundamental representation, namely (by the Frobenius-Schur indicator) a skew-symmetric bi-linear form: $$ \varepsilon = \left(...
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SU(2) invariant structures and the Frobenius Schur indicator

For $SU(2)$, the fundamental representation is a quaternionic representation. Which means there is a preserved skew symmetric form, written as a matrix: $$ \varepsilon = \left(\begin{array}{cc} 0 &...
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Is there a name for this 'splitting' of an orthogonal structure into unitary and symplectic structures?

Suppose we have a (non-trivial) representation of some special orthogonal group $SO(p,q)$ over a real vector space $V$, I.e. the action of elements of $SO$ leave invariant a non-degenerate symmetric ...
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The trace form is the unique non-deg. symmetric bilinear form such that ...

Let $L \vert K$ be a field extension. Recall the various definitions of the trace map $Tr_{L\vert K}:L\mapsto K$ (via matrices, minimal polynomials, field embeddings). With that, one gets the trace ...
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Isomorphism between a vectorial space and its dual using bilinear forms.

If $g$ is a non degenerate symmetric bilinear form over a v.s. $V$ of finite dimension I want to find an isomorphism between $V$ and its dual in the more natural way possible. I thought about defining ...
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Lattice embeddings in $\mathbb{Z}^n$

I have a certain class of lattices, and given $\Lambda$ in this class I would like to find obstructions to the existence of a finite-index lattice embedding $\Lambda \rightarrow \mathbb{Z}^n$ for some ...
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Number of solutions in a finite field corresponding to one-dimensional isotropic subspaces

Let $q$ be a prime power and define the bilinear form $$((x_1, x_2, x_3),(y_1, y_2, y_3)) \mapsto x_1 y_3^q + x_2 y_2^q +x_3 y_1^q$$ on the three-dimensional vector space $V$ over the finite field $F_{...
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What allows a bilinear form (output: field element) to also work as a linear map (output: vector)?

A bilinear form $B: V × V → K$, when the inputs are 2 vectors, has 1 element of their field as output. A linear map $L: V → W$, when the input is 1 vector, has 1 vector as output. I have seen cases ...
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Is the bilinear map on a tensor product nondegenerate?

In this question, all rings are unital commutative, and all modules are unital (i.e., $1\cdot a=a$ for all module element $a$). Definition of Inner Products Let $M$ be an $R$-module. An inner product (...
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The cube length of a vector

The form $$ {\bf r} \cdot {\bf r} = \left|{\bf r}\right|^2 = \sum_{i,j} r_i r_j {\bf e}_i \cdot {\bf e}_j = \sum_{i,j} r_i r_j \delta_{ij} = \sum_i r_i r_i $$ is defined via a bilinear form that takes ...
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Do all bilinear forms have a signature? [closed]

If yes: starting from any arbitrary bilinear form, what is the algorithm to calculate its signature? If no: what are the conditions necessary for a bilinear form in order to have a signature? Is there ...
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Calculate the basis of a bilineal form with basis change in an euclidean space

I have the following problem: f is a bilineal form in euclidean space $$ f\left(\begin{pmatrix} x_1\\x_2\\x_3 \end{pmatrix},\begin{pmatrix} y_1\\y_2\\y_3 \end{pmatrix} \right) $$ and its matrix ...
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I need a reference : A non-degenerate symmetric bilinear form on a finite k-dimensional space U over the two element field falls into two types.

Let $\mathbb {F}$ be a field, $U$ be a vector space over $\mathbb{F}$ and $\varphi$ be a bilinear form over $U$. We recall that this form is symmetric if $\varphi(x,y) = \varphi(y,x)$ for all $x,y\in ...
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Is this linear mapping positive definite, symmetric, bilinear?

$V=\mathbb{R}^2$ and let be$s:V \times V \rightarrow \mathbb{R^2} $ and $ s(x,y)=2x_1y_2+2x_2y_2$ Well I found out that the matrix is: $$\begin{pmatrix} 0 & 2\\ 0 & 2 \end{pmatrix}$$ i) It's ...
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Is nondegeneracy of a bilinear form equivalent for both slots?

According to Wikipedia, a bilinear form $B$ on a (possibly infinite-dimensional) vector space $V$ is defined as nondegenerate if the map $$B^\flat:V\ni x\mapsto B(x,\cdot)\in V^*$$ is an isomorphism ...
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Completing squares

We want to find when a quadratic form depending on a parameter $\alpha$ on $\mathbb{R}^3$ is an inner product. It is quite obvious and quick to deal with the problem by using Sylvester, BUT we want to ...
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Are "standard" and "Hermitian" quadratic forms sub-cases of the same object?

Let $V$ be a vector space over a field $k$. The definition of quadratic form on nLab is a map $q:V\to k$ such that $q(tv)=t^2q(v)$ for all $t\in k,v\in V$, and such that the map $(v,w)\mapsto q(v+w)-q(...
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