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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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How do I generalize the dot product (bilinear form) in spherical coordinates?

In cartesian coordinates, the unit vectors $\{u_x, u_y, u_z\}$ are universal. That is, $u_x(x, y, z)$ is constant and so on for the rest of them. Because of that, the dot product $\langle v | w \...
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I have this matrix. It is a Gram matrix of some bilinear form on the R^n.

(1 1/2 1/3 1/4 ... 1/(n-1) 1/n) (1/2 1/3 1/4 1/5 ... 1/n 1/(n+1)) (1/3 1/4 1/5 1/6 ... 1/(n+1) 1/(n+2)) (1/4 1/5 1/6 1/7 ... 1/(n+2) 1/(n+3)) (...................................) (1/(n-3) 1/(n-2) 1/(...
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prove a bilinear form is nondegenerate

In an exercice, I have the following: $B(M,N)=\frac{1}{2}Tr(M\tilde{N})$ where $B:M_2(\Bbb C)^2 \to \Bbb C$ and $\tilde{N}=^t(com N)$ Prove B is nondegenerate. I tried to prove that $M\in M_2(\Bbb ...
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No canonical correspondance between bilinear forms $b:V\times V \rightarrow \mathbb{R}$ and linear forms $\hat{b}:V\otimes V \rightarrow \mathbb{R}$?

In the wiki on bilinear forms, the universal product says that to each bilnear map $b : V\times V \rightarrow \mathbb{R}$ we can associate a linear map $\hat{b} : V\otimes V \rightarrow \mathbb{R}$, ...
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1answer
39 views

Integral inequality issue

Given that the bilinear form on the set $$V:= \{v\in C^2[0,1] v(0)=0=v(1)\}$$ is defined as $$[u,v]=\int_0^1 [pu'v'+quv]dx,$$ where $p\in C^1[0,1], p(x)\ge p_0>0, q(x)\ge 0, q\in C[0,1]$, I want ...
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1answer
48 views

When the characteristic of the field is not two, we can always find an orthogonal basis.

Let $ V $ be a $n$-dimensional vector space over the finite field $\mathbb F_q$, with $ \operatorname{Char}\mathbb F_q\ne 2 $. Show that for every symmetric bilinear form $B(\cdot,\cdot)$ on $V$, ...
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1answer
14 views

How to show a real matrix $A$ belongs to Indefinite Orthogonal Group $O(n;k)$?

I want to show that an $(n + k) \times (n + k)$ real matrix $A$ belongs to $O(n;k)$ iff $gA^Tg = A^{-1}$. I know that for all $\vec{x}, \vec{y} \in \mathbb{R}^{n+k}$, and for the matrix $$ g = \...
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Compact sympletic group

I was reading Brian C hall book on Lie algebra In that I come across following I had following with me But form this I can not conclude highlighted text . Please Help me . Any Help will be ...
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Question regarding Sylvesters law of inertia

In Sylvesters law of inertia it's written that Let $H$ be a bilinear form on a finite dimensional vector space $V$. Then the number of positive diagonal entries and the number of negative diagonal ...
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25 views

Showing that invertibility implies coercivity

I'm trying to proof the following: Let $f$ be a continous sesquilinear form on a Hilbertspace $H$ and let $A: H \to H$ be its dedicated Operator such that $$f(u,v) = \langle u, Av \rangle_H \;\;\; \...
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37 views

Symmetry of inner product

Peter J. Cameron's "Notes on Linear Algebra" defines An inner product on a real vector space $V$ is a function $b: V \times V \to R$ satisfying b is bilinear (that is, b is linear in the first ...
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18 views

Bilinear Maps and their relationships with dual bases

I have a theorem without proof. I have searched many books and tried on myself, but i still dont have the solution. Let M and N F-vector spaces, T be a base of M ,S be a base of N such that dimension ...
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Some questions about bilinear forms and their non-degeneracy.

I was reading this post about bilinear maps and it raised some questions in my mind, for which I am not so sure about the answers. It is proven there that for any symmetric bilinear form, it is ...
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48 views

A question on linear integral equation about non degenerated bilinear form

Let $X$ be a Banach Space , $X\subseteq H,\bar{X}=H$,where $H$ is a Hilbert space $i:=X\to H$ defined by $i(x)=x$ and is continuous. Define $\langle x,y\rangle=\langle ix,iy\rangle$ then $\langle X,X,...
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27 views

Bilinear objective functions

Are all bilinear optimization problems considered to be non-convex and non-concave? I am trying to get a counterexample if that is not true.
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1answer
53 views

Quadratic Form as Sum of Squares

I’ve been trying to prove the following: Let $k$ be an algebraically closed field with $\text{char}(k)\neq2$, and let $Q$ be a non-singular quadratic form on $k^n$. Show that for some choice of ...
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1answer
58 views

If $A$ is symmetric, why is $ x^TA^Ty=(x^TA^Ty)^T $ obvious?

The question has emerged when I read the proof of the following Theorem: Let $V$ be a $\mathbb{R}$-vector-space, with $\dim(V)=n< \infty, C=(c_{1},...,c_{n}) $ a$ $ Basis of $V$, and $B$ a ...
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1answer
40 views

$B: \mathbb R\times \mathbb R \to \mathbb R$ be the function $B(a,b) = ab $. Which of the following is true?

$B: \mathbb R\times \mathbb R \to \mathbb R$ be the function $B(a,b) = ab $. Which of the following is true? 1) $B$ is a linear transformation 2) $B$ is a positive definite bi-linear form 3)$...
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$b(v,w)$ non-degenerated implies $\hat{b}(v,w)=b(w,v)$ is non-degenerated?$

If the bilinear form $b(v,w)$ is non degenerated (meaning $\forall v, b_v(w)$ is isomorphism, does this imply that $\hat{b}(v,w)=b(w,v)$ is non-degenerated?
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14 views

bilinear form on some vector space (not necessarely finite dimension).

Let $\psi$ be a bilinear form. and consider : $\psi_L :V\rightarrow V^*$, such that $v \mapsto (f : w \mapsto b(v,w))$ $\psi_R :V\rightarrow V^*$, such that $v \mapsto (g : w \mapsto b(w,v))$ Is it ...
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1answer
17 views

Evaluating independence of vectors using Bilinear Forms

Let $f : U × U → \mathbb{R}$ be a bilinear form such that $f (u, u) > 0$ and $f (v, v) < 0$ for some $u, v \in U$. I would like to show that $u, v$ are linearly independent. We have that $u , v$...
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1answer
36 views

Completion of a subspace of a Hilbert space with respect to a positive semi-definite symmetric bilinear form

Let $H$ be a $\mathbb R$-Hilbert space $\mathcal E$ be a positive semi-definite symmetric bilinear form on a dense subspace $\mathcal A_0$ of $H$ and $$\mathcal E(f):=\mathcal E(f,f)\;\;\;\text{for }...
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1answer
65 views

How can we show that this nonnegative symmetric bilinear form is closable?

Let $(E,\mathcal E)$ be a measurable space $\mu$ be a measure on $(E,\mathcal E)$ and $$\mu f:=\int f\:{\rm d}\mu$$ for Borel measurable $f:\mathbb R\to\mathbb R$ with $f\ge0$ or $\mu|f|<\infty$ $\...
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88 views

In what sense are Minkowski spaces with $(1,3)$ and $(3,1)$ signature isomorphic?

There is no isometry between a $(1,3)$-signatured and a $(3,1)$-signatured Minkowski space, but in spite of this, they "look like" the same, for example, they have the same light cone. Is there any ...
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18 views

Bilinear form on tensor product restricted to direct summands

Let $\mathfrak{g}$ be a complex semi simple Lie algebra. Then $\mathfrak{g}$ is equipped with a canonical $\mathfrak{g}$-invariant non-degenerate bilinear form $\beta$. Now this gives a $\mathfrak{g}$-...
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Tensor Products for Bilinear Optimization?

I'm not very familiar with tensor products, but I recently read in a math textbook that " Tensor products are very important in algebra. They reduce the study of bilinear maps to the study of linear ...
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36 views

Exercise about finding an orthonormal basis

Consider the vector space $P=\mathbb{R}[x]_{\le3}$ with $*:P\times P \to \mathbb{R}$ defined by $f*g=\int_{-1}^{1}f(t)g(t)dt$. Prove that * is a symmetric bilinear form positive definite. ...
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1answer
25 views

Simplifying equation

I'm looking at answers to a question and I see these two lines of the equation. How is they're getting the -0.5095 on the bottom? eqn
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Relationship between anisotropic and negative/positive definite

Let $K$ be an ordered field, and $(V, <->)$ a non-degen. bilinear space, where $<->$ is a bilinear form. Determine whether the following statement is true or false: $(V,<->)$ is ...
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Schubert Cell Structure for some variety with prescribed bilinear form

We know the construction of Schubert cell complex structure for the case of Grassmann manifold $Gr_k(\mathbb{C}^n)$ i.e variety of all k- dimensional subspaces $V$ of $\mathbb{C}^n$. Consider that ...
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1answer
23 views

Equivalent Conditions of Nondegenerate Bilinear Forms and the Gram Matrix

One can often find the following theorem describing equivalent conditions for non degenerate bilinear forms. $\textbf{Theorem 1}$: Let $V$ be a vector space over the field $\mathbb{F}$ equipped with ...
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2answers
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Find invertible matrix $Q$

From the book "Bilinear forms and their matrices" by Prof. Joel Kamnitzer: Consider: $$ A=\pmatrix{0 & 4 \\ 4 & 2} $$ After doing simultaneous row and column operations we reach: $$ Q^...
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1answer
49 views

Prove that a bilinear form is nondegenerate

Let $n$ be a positive integer, $V=\mathbb{C}^{n\times n}$ and $$f(A,B)=n\mathrm{Tr}(AB) - \mathrm{Tr}(A)\mathrm{Tr}(B).$$ Let $$W=\{ A\in V\, |\, \mathrm{Tr}(A)=0 \} $$ and let $f_1=f|_{W}$. Prove ...
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1answer
22 views

Limit of non-degenerate biliniear forms

Let $X$ be a Banach space (in my specific case I have $X=C_b(\mathbb{R})$) and let $\{B_n\}_{n\geq 1}$ be a sequence of biliniear forms $B_n:X\times X\rightarrow \mathbb{C}$, which are non-degenerate ...
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How to show bilinearity

Given a Coxeter group $\varGamma =\langle \rho_0, \rho_1, \ldots, \rho_{n - 1}\rangle$ which at least satisfy the relation $(\rho_i\rho_j)^{p_{ij}}=1, \ \ 0\leq i, j \leq n - 1,$ where $p_{ii}=1$ and $...
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A Symmetric bilinear variation for self congruence?

--edited-- May I know the group or condition that satisfies $$S^T A S = A$$ possibly when $A_{n\times n}$ is symmetric? Meanwhile, below is an example for $S_{3\times 3}$. MATLAB code: ...
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1answer
21 views

For a bounded linear functional $l(v) := (f,v)_\Omega$, do we have $\|l\| \le \|f\|$?

Suppose we have a Poisson equation $-\Delta u = f$ on $\Omega$ and we want to derive its weak formulation, so we multiply it by an arbitrary test function $\forall v \in H^1_0$ and then take integral ...
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1answer
41 views

Process of finding orthonormal basis of given bilinear form

Let $V$ be vector space of $n\times n$ matrices . $\langle A,B \rangle = {\rm tr}(A^T B)$ I wanted to find orthonormal basis of it. I know that if there small say 3 dimension vector say then I ...
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How to show that every symmetric nonsingular complex matrix can be written as $A=P^ T P$ form

I know that if we have a positive definite matrix then we can show that A=$P^T P$ But here I had given to prove for a complex matrix which needs to be hermitian. It is not duplicate AS My question ...
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1answer
41 views

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$ I am studying now bilinear form .I wanted to prove above theorem. I ...
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25 views

Is there a canonical isomorphism between $V\otimes_F S$ and $V_S\oplus V_S$?

Let $E/F$ be a CM extension of number fields (i.e., $F$ is totally real and $E/F$ is a totally imaginary quadratic extension). I assume $E=F[\alpha]$ with $\alpha\in E\backslash F$ such that $\alpha^2\...
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28 views

Reduced equation of quadric surfaces

Given the following quadric surfaces: Classify the quadric surface. Find its reduced equation. Find the equation of the axes on which it takes its reduced form. The quadric surfaces are: (1) $3x^2 +...
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Which is the markovian process associated to the Dirichlet form given by the inner product?

I know that there is a relation between Dirichlet forms and Markov processes. In particular, for every regular Dirichlet form, there exist a Markovian process associated. I would like to understand ...
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1answer
28 views

bilinear transformation $\phi U\times V\to W$ such that $Im(\phi)=\{\phi(u,v): u\in U, v\in V\}$ is not a subspace of $W$

Find a bilinear transformation $\phi U\times V\to W$ such that $Im(\phi)=\{\phi(u,v): u\in U, v\in V\}$ is not a subspace of $W$ I truly don't have an idea otherwise to brute force lots of tries and ...
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4answers
100 views

Let ${^ts}:E\to F$ be the transpose of $s:F\to E$. Show that $\text{Im}(s)\cap \ker (\,^ts)=\{0_E\}$.

Let $s\in \mathcal{L}(F,E)$ $$\displaystyle F \overset{s}{\longrightarrow} E\overset{^ts}{\longrightarrow} F$$ I spent two days to show that :$$\text{Im}(s)\cap \ker (\,^ts)=\{0_E\}\qquad \tag{...
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1answer
45 views

bilinear form only positive or negative

Let $f$ be a definite, symmetric bilinear form. Show that $f$ is positive or negative. Consider the quadratic form $q(x,y,z)=x^2+y^2-z^2$, then the polar form assocaited to $q$ is symmetric, ...
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32 views

Bilinear form can be written as sum of hermitian and symmetric C bilinear form

Suppose $g : E \times E \rightarrow \mathbb{C}$ is $\mathbb{R} $ bilinear.Show that there exists a hermitian form $h$ on $E$ and a symmetric $\mathbb{C}$-bilinear form $\psi$ on $E$ such that $2ig= h+...
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22 views

To show orthogonal basis exist consisting common eigenvectors

Let $h$ be a positive definite hermitian form on $E$ and $A,B: E\rightarrow E$ be two hermitian endomorphisms which commute, $AB = BA$. Prove that there exists a orthogonal basis for $E$ consisting of ...
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1answer
34 views

Hermitian form can be written as sum of symmetric and alternating form

Let $h:E\times E \rightarrow \mathbb{C}$ be a hermitian form.Consider the real and imaginary part of $h$ : $h(x,y)= g(x,y)+ if(x,y)$ Prove that $g$ and $f$ are bilinear,$g$ is ...
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43 views

Ways to check positive definiteness of bilinear form

For each real number $\alpha$, we define the bilinear form $F_{\alpha}:\mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R} $ by $\displaystyle F_{\alpha}(((x_1, x_2, x_3), (y_1, y_2, y_3)) = ...