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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signature of a bilinear form

This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8. Lemma: let $E$ be an n-dimensional real vector space endowed with a ...
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bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
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Positive definiteness of a bilinear form implies symmetry?

In the Wikipeda article about positive definite bilinear forms, there is the line It turns out that the matrix $M$ is positive definite if and only if it is symmetric and its quadratic form is a ...
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Sesquilinear Forms

I was trying to solve some exercises related to sesquilinear forms: Let V be a C-vector space (C - complex numbers) Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on V is a vector ...
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Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad ...
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If $W$ is proper subspace of a real finite dimensional $V$, $\exists v\in V $ s.t. $H(w,v)=0 \forall w \in W$

$H(x,y)$ is a bilinear form. I have tried to do something similar to standard inner product in $\Bbb{R}$ such as Gram-Schmidt process, orthogonal complement...But they don't work since we don't know ...
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Orthogonal projection of skew-symmetric form

It is a question from the book Algebra by Michael Artin: 8.8.2 Let W be a subspace on which a real skew-symmetric form is nondegenerate. Find a formula for the orthogonal projection $\pi:V\to W$
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Prove the positive definiteness of Hilbert matrix

This is so called Hilbert matrix which is known as a poorly conditioned matrix. $$ A = \left(\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} & ... & \frac{1}{n} \\ ...
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There is no positive definite bilinear form on $V$

Let $V$ be a $\mathbb C$-vector space, $\dim V>1$. Then for any bilinear form $\phi$ there is $v\in V$ s.t. $\phi(v,v)=0$ (so there is no positiv definite bilinear form on $V$) If one takes ...
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Perfect Pairing, non-degeneracy and dimension.

On this wikipedia entry https://en.wikipedia.org/wiki/Bilinear_form#Different_spaces it tells us that if $B: V \times W \to K$ is a bilinear map, then In finite dimensions, [a perfect pairing] is ...
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Orthogonal complement of orthogonal complement of U equals U

Let $V$ be of finite dimension over some field $F$. Let $ξ\in T_2(v)$ symetric bilinear form. Let $U\subset V$. suppose $ξ|_U$ is not degenerate, is it necessarily $(U^⊥)^⊥=U$ ? my approach: I ...
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Weierstrass theory of elementary divisors

I'm taking a history of algebra course and I can't seem to figure out Karl Weierstrass' Theory of Elementary Divisors. It says he began by considering two quadratic forms $\phi = \sum_{i,j=1}^n ...
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29 views

Find the orthogonal complement with respect to $\phi$

Given is a symmetric bilinear form $\phi:\mathbb R^4\times\mathbb R^4\to \mathbb R$ with $\phi(v,w)=v^{T}Aw$ and ...
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52 views

Hermitian Pairings from Positive Functionals

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, ...
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Root systems and the possible angles between roots.

On page 4 of these notes by John Dusel (http://math.ucr.edu/~jmd/Root_Systems.pdf) it reminds us that if we have a symmetric positive bilinear form (pageg 2) we can define an angle between vectors ...
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There exists another bilinear symmetric map which is a multiple of $F$

This question is a continuation of the following question. So we have $F: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ a bilinear symmetric map, and $K = \{ v \in \mathbb{R}^n \mid F(v,v) = 0 ...
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$F$ is indefinite if and only if $K$ is not a subspace

I get the following linear algebra problem in my class. Let $F: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ be bilinear and symmetric, and let $K = \{ v \in \mathbb{R}^N \mid F(v,v) = 0 \} $. ...
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Is the converse to this theorem true?

In the book that I'm reading there is this one theorem which states. Let V be a finite-dimensional vector space over a field F not of characteristic two. Then every symmetric bilinear form on V is ...
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42 views

Problem involving Bilinear Forms from Micheal Artin's book.

The question is: Let T be a linear operator on $V=R^n$ whose matrix $A$ is a real symmetricmatrix. a) Prove that $V$ is the orthogonal sum $V=(ker T)\oplus(im T)$ b) Prove that $T$ is an orthogonal ...
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signature of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ (using homology, cohomology)

I want to prove that the signature $\operatorname{sig}(M)$ of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ is nonzero. First of all, $M$ is a compact 24-dimensional manifold ...
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questions about a proof of a theorem about symmetric nondegenerate bilinear forms

Let $V$ be a finite dimensional $\mathbb{R}$-vector space ($\dim_\mathbb{R} V=n$) and $\varphi:V\times V\to\mathbb{R}$ a symmetric nondegenerate bilinear form. Then there exist subspaces $V^+$ and ...
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Bilinear (non-convex) objective and linearized (big-M) constraints

My original problem is a MIQCQP. The bilinear terms in the constraints are products of binary and continuous variables and can be linearized using big-M. The bilinear terms in the objective function ...
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Simultaneous Diagonalization of two bilinear forms

I need to diagonalize this two bilinear forms in the same basis (such that $f=I$ and $g$=diagonal matrix): $f(x,y,z)=x^2+y^2+z^2+xy-yz $ $g(x,y,z)=y^2-4xy+8xz+4yz$ I know that it is possible ...
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Checking alternating tensors on $\mathbb R^4$

I'm trying to solve the following question : Which of the following are alternating tensors in $\mathbb R^4$ and express those that are in terms of the elementary tensors on $\mathbb R^4$: ...
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Symmetric bilinear forms and (continuous) dual spaces

Let $V$ be an infinite dimensional locally compact vector space over a field $k$ (the field $k$ has the discrete topology and on $V$ we fix the linear topology ). Moreover suppose that on $V$ is ...
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Skew-symmetric non-degenerate bilinear form and $J^2=-I$ operator

Let $f:V\times V\to \Bbb R $ skew-symmetric non-degenerate bilinear form in a real vector space $V$. prove that there is operator $J:V\to\ V $ such that: $J^2=-I$ Form $\varphi:V\times V\to \Bbb R ...
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Is this a bilinear function?

I am trying to understand what kind of problem is a bilinear form. Does the following example (zz function) qualify as bilinear? Matlab code: [x1,x2]=meshgrid(-1:0.1:+1,-1:0.1:+1); zz=x1.*x2; ...
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I want to find the following matrix problem related to eigenvalue and eigenvectors

$$ A=\begin{pmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \\ \end{pmatrix} $$ $A$ has eigenvalue $ \lambda $ of multiplicative 2 and eigenvalue $ \mu $ of multiplicative $1$. ...
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Prove that if positivity of one bilinear form implies positivity of second bilinear form then they are scalar multiples.

Let $M_2, M_2 \in \mathbb{R}^{d\times d} \setminus \{ 0 \}$. Prove that if for all $x,y \in \mathbb{R}^d$ $$x^T M_1 y > 0 \implies x^T M_2 y > 0$$ then $M_2 = \lambda M_1$.
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Solving bilinear systems on a limited domain

Given a diagonal matrix $A$ with all entries $A_{i,i}\in(0,1)$ and trace equal to one $$ A\in \mathbb{R}^{k \times k} $$ and a second matrix $$ B\in \mathbb{R}^{n \times k},\,\,\,\,\,\,n\geq ...
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What do the properties of the dot product mean? [closed]

I am having trouble understanding the relevance or meaning of the properties of dot product. For example, the distribution property of dot product states: $$\vec a \cdot (\vec b+\vec c) = \vec a ...
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Proof that bilinear form in $H_0^1$ is coercive

Let $$B(u,v)=\int_I uv + \int_I u'v'$$ where $u,v\in H_0^1(I)$ for a given interval $I=[a,b]\subset\mathbb{R}$. How can I prove that the bilinear form $B$ is coercive, i.e., that $$B(u,u)\ge C\Vert ...
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Is this polynomial minimization equivalent to my original problem?

I'm trying to find two vectors $\mathbf{x}$ and $\mathbf{y}$, such that their entries satisfy a system of equations, each one in the form $$\sum_{(i,j)\in J} x_iy_j=0$$ for a given collection of $J$'s ...
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1answer
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Product of the norms of two vectors w.r.t a symmetric bilinear form

Let $V=V_{n}(q)$ be a $n$ dimensional vector space over the finite field $\mathbb{F}_{q}$ and let $(,)$ be a symmetric bilinear form on $V$. Fix $v\in V$. I would like to show that there exists a ...
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matrix trace bilinear form

I'm wondering about this problem on bilinear forms : We have $\phi : \mathbb{M_{n}(R)}*\mathbb{M_{n}(R)} \rightarrow \mathbb{R}$ $$(A,B) \rightarrow trace(AB)$$ I've proved $\phi$ is a bilinear form ...
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1answer
59 views

Null Bilinear Forms $x^T A y = 0$, where $A$ is square and full rank.

Let A be a full rank square matrix (A has no null space). When does $y^T A x = 0$ occur ? It could be that this problem is case-specific, so please find attached a document where x,y, and A take ...
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Is the matrix filled with the areas of pairwise intersections of disks in a plane always positive semidefinite?

Consider disks $s_1, \cdots, s_n$ in the plane and let $a_{ij}$ be the area of $s_i\cap s_j$. Is it true that for any real numbers $x_1,\cdots, x_n$ we have $$ \sum_{i,j=1}^n x_ix_j a_{ij} \geq 0$$ ...
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Identification of a curious bilinear form

As I was solving a linear algebra exam, a wild bilinear form $f:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ appeared: $$f(x,y)=n\cdot\sum_{i=1}^n{\left(x_i-m_x\right)\left(y_i-m_y\right)}$$ ...
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1answer
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Example of degenerate scalar product

Suppose I have a vector space $V$ over $\mathbb{R}$. A scalar product on $V$ is a symmetric bilinear form, so such that $\langle v,w\rangle = \langle w,v\rangle~ \forall v,w \in V$. To my ...
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Inner Product on $P_3(\mathbb{R})$ true/false

The question was to classify each statement as true or false with justification: Define $\langle f, g\rangle$ = $\int_0^1 f'(x)g(x) + f(x)g'(x) \,dx $. Then, $\langle , \rangle$ is an inner product ...
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$\det$ is the only multiplicative nonzero quadratic form on $\mathcal M_2(\Bbb R)$

Let $q$ a nonzero quadratic form on $\mathcal M_2(\Bbb R)$ verifying the relation $$\forall A,B\in\mathcal M_2(\Bbb R),\; q(AB)=q(A)q(B)$$ The question is to prove that $q=\det$. What I have tried ...
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43 views

Subspace $U\subset U^{\perp}$ has dimension at most $n/2$

We have $V$ a $\mathbb{R}$-vector space of finite dimension $n$ with non-degenerate bilinear form $\phi:V\times V\rightarrow \mathbb{R}$ and subspace $U$ of $V$ with $U\subset U^{\perp}$. How do I ...
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1answer
24 views

Bilinear and linear map, then zero map

If $b:V\times V\rightarrow W$ is both bilinear and linear, then it is the zero map. How do I prove this? What I tried: linear: $b((x,y)+(x',y'))=b(x,y)+b(x',y')$ and ...
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1answer
29 views

GNS construction of a weight

In the theory of quantum groups in the operator algebraic setting, one deals with weights (instead of positive linear functionals). Definition: A weight is a function $\phi $ : $A^+ \rightarrow [0, ...
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Bilinear forms defines inner product on Hilbert Space

I have difficulties understanding the reason why when I have a self adjoint linear operator $T : \mathcal{H} \rightarrow \mathcal{H}$, and know that $A\|f\|^2 \leq \langle Tf,f \rangle \leq B\|f\|^2$ ...
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Using Lagrange's diagonalization on degenerate linear forms

Let $A=\begin{pmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{pmatrix}$ be a real matrix. Find an invertible matrix $P\in M_{3}(\mathbb{R})$ such that $P^TAP$ is diagonal ...
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32 views

Bilinear maps and non-degeneracy

Let $\mathbb{G}_0$ and $\mathbb{G}_1$ be two multiplicative cyclic groups of prime order $p$, $g$ a generator of $\mathbb{G}_0$ and $e$ a bilinear map, ...
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80 views

What is the meaning of the eigenvalues of the matrix representation of a bilinear form?

Given a bilinear form $B$ on some finite-dimensional vector space $V$, we can always represent $B$ by some matrix $A$ such that $B(v,w) = [v]^TA[w]$. Thus we could associate the eigenvalues of $A$ ...
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How to show that the vectors of $ker \phi$ are linearly indipendent to other vectors in a orthogonal basis with respect to $\phi$

How can I show that, given a symmetric bilinear form in a finite dimensional vector space $V$ $\phi : V\times V \rightarrow \mathbb{R}$ then the vectors of the basis of $ker \phi$ (which I assume that ...
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2answers
38 views

A quadratic form is positive-definite iff its set of isotropic vectors is trivial

Considering a quadratic form $Q$ in a finite dimensional vector space $V$ can I say that $\mathscr{I}=\big\{ \vec{o} \big\} \iff Q $ is definite positive ? Where $\mathscr{I}$ is the isotropic ...