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.

learn more… | top users | synonyms

1
vote
2answers
50 views

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?
2
votes
1answer
46 views

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 ...
0
votes
1answer
28 views

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 = ...
0
votes
1answer
19 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 ...
0
votes
0answers
27 views

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 ...
0
votes
2answers
32 views

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 ...
1
vote
1answer
22 views

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 ...
1
vote
0answers
16 views

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, ...
2
votes
2answers
30 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 $$\lim\limits_{(h,k) \to (0,0)} \dfrac{|f(h,k)|}{|(h,k)|} = 0$$.
2
votes
0answers
50 views

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 ...
0
votes
0answers
32 views

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 $$ f(v_{i},v_{j}) = -f(v_{j},v{i}). $$ ...
1
vote
1answer
98 views

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 ...
1
vote
1answer
33 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 ...
0
votes
1answer
33 views

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 $Q$
0
votes
1answer
28 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 ...
0
votes
1answer
43 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 $ ...
0
votes
0answers
54 views

Symmetric Positive-definite matrix.

let $f:V \times V \rightarrow {R}$ not Degeneracy skew symmetric bilinear form in vector space $V$ over $R$ $\forall v,u \in V: f(u,v)=u^tAv$ , $A=-A^t$ . Let $J:V \rightarrow V$ such that $J^2=-Id$. ...
0
votes
0answers
18 views

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 ...
1
vote
1answer
29 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 ...
1
vote
2answers
48 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 ...
4
votes
1answer
60 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 ...
2
votes
4answers
189 views

Tensor product in multilinear algebra

In the book by Halmos ($FDVS$) the tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised ...
0
votes
2answers
30 views

Find an orthonormal basis of a particular bilinear form

Let $V=\mathbb{R^3}$. Find an orthonormal basis in which the bilinear form with matrix $A$: $\begin{pmatrix} 2 & -2 & 0 \\ -2 & 1 & -2 \\ 0 & -2 & 0\end{pmatrix}$ has a ...
1
vote
1answer
24 views

Condition for continuity of bilinear form

In my numeric script there is a unproved theorem, saying that a bilinear form $a \colon V\times V \to \mathbb{R}$ on a normed vector space $V$ is continuous if and only if $$|a(v,w)| \leq c \, \|v\| ...
1
vote
0answers
23 views

Centralizing a maximal flag in a symplectic group

Short version: I'm confused about maximal totally-isotropic flags versus maximal flags: do they have the same centralizer in the classical group? Let $F$ be a field, $V$ be a finite dimensional ...
0
votes
1answer
28 views

Bilinear transformation and eigenvalues

I have a proof to do and I am stuck on proving that if there exist a matrix $A$ with eigenvalue $\lambda_i$ and $B$ with eigenvalues $\mu$ such that $A = (B+I)(B-I)^{-1}$ then we have ...
1
vote
1answer
13 views

Find basis for $P_0^\perp \subset P_4$ and $\ker (f \mapsto f(0))$ in $P_4$

Let $P_n \subset \textrm{Map}(\mathbb{C},\mathbb{C})$ the space of polynomial maps $\mathbb{C} \to \mathbb{C}$ with degree $\le n$. We define $\langle f,g\rangle := \int_{-1}^1 f(t)\overline{g(t)}dt$. ...
0
votes
2answers
67 views

Diagonalizng a bilinear form

Question a. we mark $\mathbb{R}_2[x]$ as the polynomial space of degree $ \le 2$ over the real field $\mathbb{R}$. $\xi :\mathbb{R}_2[x] \times \mathbb{R}_2[x] \to \mathbb{R}$ $$\xi(q,p) = ...
0
votes
1answer
26 views

Matrices in Linear Algebra

Let: $ u: R^2 --> R^3$ be defined by: $$ u(x,y)=(x+2y, 2x-y, 2x+ 3y)$$ Give the matrix $M[u]$ in the canonical base of its definition space. This question might seem sort of stupid, but it was ...
0
votes
1answer
45 views

Show that $B$ represents an inner product.

Let $V$ be a vector space over $\mathbb{R}$ with $\dim V = n$ Let $B = (b_{ij})$ be an $n \times n$ diagonal matrix which represents a bilinear form on $V$ with respect to a basis of $V$ , ...
4
votes
1answer
93 views

Find basis such that bilinear forms attain normal form

Let $g :=$ symmetric bilinear form : $g(v,w) := \omega(v,Jw)$ with $J \in \textrm{O}(V,g)$ and $\omega := $ skew-symmetric bilinear form : $\omega(v,w) := g(v,Jw)$ with $J \in \textrm{Sp}(v,\omega)$. ...
1
vote
1answer
47 views

Norm of an operator defined on sequence spaces

Consider the sequence space $\ell_r$ defined by $$\ell_r=\left\{x=(x_n)_{n=1}^{\infty}:x_n\in\mathbb{R}\text{ and }\sum_{n=1}^{\infty}|x_n|^r<\infty\right\}.$$ Let $2\leq p,q<\infty$ such that ...
2
votes
0answers
45 views

Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
1
vote
0answers
32 views

Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...
4
votes
1answer
100 views

Show isomorphism $W_1 \hookrightarrow V \twoheadrightarrow W_2$

Let $\langle , \rangle$ be a non-degenerate bilinear form with the signature $(p,q)$ on a real vectorspace $V$ and $W_1, W_2$ subspaces, such that the restriction $\langle , \rangle |_{W_i}$ is ...
1
vote
1answer
27 views

Show skew-symmetric, non-degenerate bilinear form $((a, \varphi),(b, \psi)) \mapsto \langle(a, \varphi),(b, \psi) \rangle := \varphi(b)-\psi(a)$

Let $W$ be a finite dimensional $K$ vector space and $W^*$ its dual space. For $V := W \oplus W^*$ the mapping $$ V \times V \to K,((a, \varphi),(b, \psi)) \mapsto \langle(a, \varphi),(b, \psi) ...
2
votes
0answers
36 views

Show $\langle , \rangle |_W$ non degenerate $\implies$ $\langle , \rangle |_{W^\perp}$ non degenerate

Let $W \subset V$ be a subspace and $\dim V < \infty$. If $\langle , \rangle$ and the restriction $\langle , \rangle |_W$ are non degenerate, then $\langle , \rangle |_{W^\perp}$ is non degenerate ...
0
votes
1answer
33 views

Very quick question on sesquilinear forms seen as bilinear maps.

This is a very quick question on sesquilinear forms when seen as bilinear maps. Let $V$ be a complex vector space, a sesquilinear map (or conjugate-linear in the first variable and linear in the ...
1
vote
2answers
40 views

Symmetric matrix congruency

There is a sentense that says that every symmetric matrix is congruent to a diagonal matrix. I've been trying to find the congruent matrix and the transition matrix for the following: $$ ...
1
vote
1answer
11 views

If a symmetric bi-liear form is positive, then $a_{11}\cdot a_{nn}>a_{n1}\cdot a_{1n}$

I need to prove that if a bi-linear form is symmetric and positive, then it's representative matrix $A=(a_{ij})_{i,j=1}^n$ satisfies: $$a_{11}\cdot a_{nn} > a_{n1}\cdot a_{1n}$$ I've tried for ...
0
votes
0answers
37 views

Generalisation of Gramian determinant

i'm wondering about those facts of basic linear algebra: if you have $n$ vectors $x_1,...,x_n \in \mathbb{R}^n$, you can easily test their linear dependance by computing their Gramian Matrix $M$ whose ...
1
vote
2answers
40 views

Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
1
vote
0answers
46 views

Linear systems and bilinear forms

Given a general linear system $Ax = y$ and the bilinear form $z(x,y) = y^T Ax$, what are the links between these two mathematical objects? Thanks. EDIT: Original question is too general and ...
0
votes
0answers
15 views

Positive definite integral quadratic form with minimal orthogonal group?

Are there explicit examples in every rank of positive definite integral quadratic forms with orthogonal group $\pm 1$?
5
votes
0answers
68 views

Closed orbits of complete flags in $\mathbb{C}^n$

Let $B$ be a symmetric (or antisymmetric) non-degenerate bilinear form on $\mathbb{C}^n$ and let $G$ be the associated group of automorphisms $O(n)$ (resp. $Sp(n)$). What can we say about the ...
1
vote
1answer
36 views

Symmetric bilinear forms in characteristic 2

This is a homework question: Prove that: In field $K$ of characteristic $2$, for symmetric bilinear forms on $K^2$, there exist a basis where the matrix of the bilinear form is either diagonal or ...
4
votes
0answers
52 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
0
votes
0answers
22 views

Bilinear form on the space of smooth complex valued functions.

Let $G$ be a Lie group and $h$ be the Hermitian bilinear form on smooth complex valued functions then how can we define bilinear form on the space of smooth complex valued functions.
2
votes
1answer
60 views

Proof: $g\mbox{ is self-adjoint endomorphism}\leftrightarrow \forall i,j \in \{1,2,…,n\}(f(g(b_i),b_j)=f(b_i,g(b_j)))$

let $g \in End_K(E)$, and $f: (E \times E)\to \Bbb{R}^1$ a symmetric bilinear form positive definite, and $(b_1,b_2,...,b_n )$ a basis, then $$f \mbox{ is self-adjoint endomorphism } \leftrightarrow ...
0
votes
0answers
45 views

Definition of self-adjoint endomorphism

let $f \in End_K(E)$, and $g: (E \times E)\to \Bbb{R}^1$ a symmetric bilinear form positive definite, $f$ is self-adjoint endomorphism if $$\forall v,w \in E(g(f(v),w)=g(v,f(w)))$$ It is correct? ...