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
1answer
16 views

Conservation of bilinear forms and conjugation

Let $\omega,\omega'$ be non-degenerate skew-symmetric bilinear forms on $V$, a vector space over $\Bbb{C}$, preserved by $G,G' \subset GL(V)$ respectively. Must there be an element $\gamma \in GL(V)$ ...
0
votes
1answer
38 views

Reflexive bilinear forms.

Let $V$ be a vector space and $B: V \times V \to \Bbb R$ be a bilinear form. Usually, I see books defining that if $B$ is symmetric, vectors ${\bf u},{\bf v} \in V$ are $B$-orthogonal if $B({\bf ...
1
vote
1answer
30 views

Clarification about finding a bilinear form's matrix

Say $V:=M_{2\times 3}(\mathbb{R})$ and let $f:V\times V \to \mathbb{R}$ be defined $f(X,Y)=Tr(X^TAY)$ for $A=\begin{pmatrix} 1 & 2 \\ 3 & 4\\ \end{pmatrix} \, $. So I want to find the the ...
1
vote
2answers
23 views

Rank of a Decomposable Tensor

I'm independently studying Stephen Roman's Advanced Linear Algebra, and I came across a line of reasoning that appears obvious but that I don't understand, and was hoping someone might help me ...
1
vote
1answer
22 views

Kernel of operator composed with adjoint coincides with original kernel.

Let $V$ be a finite-dimensional vector space, and let $\langle\cdot,\cdot\rangle$ be a nondegenerate bilinear form on $V$. If $T$ is a linear operator of $V$, does it follow that $T$ and $T^*T$ ...
2
votes
3answers
77 views

Prove that there is a symmetric matrix B, such that BX=Y

Let $X,Y$ be two vectors in ${\mathbb C}^n$ and assume that $X≠0$. Prove that there is a symmetric matrix $B$ such that $BX=Y$.
1
vote
0answers
33 views

Coercivity of a sesquilinear form on a Hilbert space

Given two Hilbert Spaces $(V,||\cdot||)$ and $(H,|\cdot|)$ with the compact inclusion $V\hookrightarrow H$ and a sesquilinear form $a(\cdot,\cdot)$ on $V$ such that: $\bf (i)$ $Re\ a(u,u)\geq 0\ ...
0
votes
0answers
18 views

Existence of a Subspace on which a Bilinear Form is Non-Degenerate

Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a bilinear form on $V$. It is known that if there exists a subspace $W$ of $V$ such that $f$ is non-degenerate on $W$, then ...
0
votes
0answers
30 views

Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
1
vote
1answer
15 views

Lagrange Method for Presenting Bilinear form as sum of squares

I have the following question in my assignment which I'm having a hard time solving. For the following bilinear form, present find a digonal form (diagonal matrix form): What I thought to do at ...
1
vote
2answers
33 views

Bilinear form and cross product in hyperbolic geometry

I'm reading Patrick J. Ryan's Euclidean and non-Euclidean geometry, page 152. There is a bilinear form defined by $b\left( {x,y} \right) = {x_1}{y_1} + {x_2}{y_2} - {x_3}{y_3}$ on ${\mathbb{R}^3}$ and ...
0
votes
0answers
31 views

Bilinear Form Non-Degenrate on a Subspace

I am trying to prove the following standard result: Let $V$ be a finite dimensional vector space over a field $F$ and $f:V\times V\to F$ be a symmetric bilinear form on $V$. Let $W$ be a subspace ...
1
vote
2answers
36 views

Symmetric bilinear forms, quadratic forms and matrices

I have computed B=$ \left( \begin{array}{ccc} 0 & 4 & -1 \\ 4 & 2 & 3 \\ -1 & 3 & 1 \end{array} \right) $ Is this correct? If so, even though I may have achieved the correct ...
0
votes
1answer
30 views

Eigenvectors, bilinear forms and orthonormal bases

I have calculated (a) to be $(1,-2,2)^t, (-2,1,2)^t, (2,2,1)^t$. For (b) I have made all of these of unit length ie taken 1/3 of each vector. I have verified these are orthonormal by checking ...
0
votes
0answers
32 views

Computing the basis of a bilinear form

I am struggling to answer part (a), I don't see how I get this into matrix form? (b) is straightforward. For (c) I must use $C^t*B*C$ where B is the matrix composed of the basis vectors, so this is ...
0
votes
0answers
66 views

When are sesquilinear forms actual inner products?

I read it was enough (and necessary) to have $\overline A^T=A$ and $A$ non-singular, for a sesqui-linear form on $\mathbb{C}^n$ to be an actual inner product. (Here $A$ is a matrix for the ...
4
votes
2answers
113 views

Is this following bilinear form coercive?

First of all I want to mention that this is homework, so don't spoil it and reveal all the answer. just some guidenss :) Let $H$ be a Hilbert space, $T:H\rightarrow H$ a bounded linear operator for ...
0
votes
1answer
39 views

Two versions of Lax-Milgram theorem

I'm having some troubles differentiating between two versions of Lax-Milgram theorem, one shown in my class and one that I saw is common on the internet. Let $H$ be hilbert space, $B$ bilinear form ...
1
vote
1answer
48 views

Diagonalisability of Self-Adjoint Operators for Non-Symmetric Metrics

Let $V$ be a finite dimensional vector space and $(\cdot,\cdot)$ a non-degenerate bilinear form. When $(\cdot,\cdot)$ is symmetric, every self-adjoint operator on $V$ is diagonalisable. What happens ...
0
votes
2answers
46 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
1
vote
1answer
51 views

Conjecture that $A^{T}BA = ABA^{T}$ for any symmetric matrix $B$ in $\mathbb{R}^n$

While trying to understand the Kalman filter, and by experimentation with Python I came up with the conjecture in the title. First of all is it true? Second, if it is, how can I prove this? I would ...
0
votes
0answers
23 views

If I define a map on basis elements from $A\times A$ to $A$, can I extend this to a bilinear map?

This may seem a silly question but I have a map $f:A\times A\to A$ where $A$ is the dual of the augmentation ideal of $\mathbf{Z}[C_n]$ (so it has a $\mathbf{Z}$-basis). This map is defined for the ...
1
vote
0answers
21 views

In a vector space $V$, If $v^tAw = v^tBw$ for all $v,w \in V$, does it imply $A=B$?

OK. So, basically this question came up when I was trying to solve a homework question about how the matrix representation of a bi-linear form on $V$ changes if we change the basis on $V$. Let's say ...
0
votes
0answers
18 views

A dimension relation on non-degenerate bilinear forms

Let $E$ be a finite dimensional $k$ -vector space and $F$ a subspace. Let $f:E\times E \to k$ be a bilinear form which is non-degenerate on its restriction to $F$. Is it true that ${\rm dim } F + {\rm ...
2
votes
1answer
43 views

Average value of a bilinear map on a Euclidean sphere

Let $(V, g = \langle \cdot, \cdot \rangle)$ be a Euclidean vector space and $B : V \times V \to \mathbb{R}$ be a symmetric bilinear form. I would like to know if something like this is true: ...
0
votes
0answers
66 views

Explicit Isomorphism between Vector Spaces

Let $V$ and $W$ be two finite dimensional spaces. I want to show that I have a canonical isomorphism from the space of bilinear forms $\mathcal{B}= \left\lbrace B: V^* \times W^* \rightarrow ...
0
votes
1answer
20 views

Non-singular bilinear forms over a vector space.

I have a field $k$ and a finite dimensional $k$-vector space $E$. Let $f$ be a symmetric $k$- bilinear form on $E$. I define $f$ to be non-degenerate if $f(x,y)=0$ $\forall y\in E$ implies $x=0$. I ...
0
votes
0answers
14 views

Notation question about scalar products and bilinear forms

Quick notation question. Is it necessary to distinguish between a scalar product and say a bilinear form $A: V \times V^* \rightarrow \mathbb{R}^n$. Would it be recommended that say you define ...
5
votes
1answer
82 views

Conic section: What is the coordinate matrix of its bilinear form?

Given is the conic section $x^2 + xy + y^2 + 2x +3y - 3 = 0$. I need to find the coordinate matrix $M_\beta(s)$ of the bilinear form $s: \mathbb{R}^2 \times \mathbb{R}^2 -> \mathbb{R}$. I read ...
-1
votes
1answer
27 views

Calculating the signature of matrix A?

given is a symmetrical bilinearform s that has the following matrix: $A = M_\beta(s) = \begin{pmatrix} -3&0&-1\\0&-3&0\\-1&0&-1\end{pmatrix}$ I have to calculate the ...
1
vote
1answer
43 views

Linear transformations in bilinear form

Be $f:V \times V \to F$ a bilinear pattern and $V$ of finite dimension. Is it correct that for every linear transformation $T:V \to V$ exists another linear transformation $T':V \to V$ for which: ...
0
votes
1answer
32 views

Understanding a bilinear form problem from Greub's Multilinear Algebra

I read the following problem from exercise sets of Greub's Multilinear Algebra, Chapter I, Sec. 1 Let $E$, $E^*$ be a pair of dual spaces and assume that $\mathit{\Phi}:E^{*}\times E\to\Gamma$ is ...
1
vote
1answer
62 views

If $A \succeq B$ is it true that $B^{-1} \succeq A^{-1}$

If $A$ and $B$ are two positive definite matrices such that $A - B$ is nonnegative definite, is it true that $B^{-1} - A^{-1}$ is positive definite? The doubt came to me when working with confidence ...
3
votes
1answer
74 views

Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
2
votes
0answers
54 views

Linear Algebra, Quadric form

Question from an exercise Let $V$ be a vector space over a field $F$ with $charF\neq2$. If $\varphi,\psi\in V^{\vee}$ are linear functionals, we will define $\varphi\cdot\psi \colon V \rightarrow ...
4
votes
1answer
237 views

Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
3
votes
2answers
70 views

$\ker(A)=\text{Im}(A^*)^\perp$

How do I show that $\ker(A)=\text{Im}(A^*)^\perp$ for any square matrix $A$. I have done this problem before with the linear operator $T$ on a hermitian space but I can't seem to apply what I have ...
1
vote
2answers
56 views

Linear Algebra, Quadric Form, Bilinear Form

I have a question from an exercise. So $V$ is a vector space with quadric form $q:V\to \mathbb{R}$ . I have to prove that if the exists $u$, $v$ in $V$ such that $q(v)>0$ and $q(u)<0$ then ...
1
vote
2answers
63 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
56 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
40 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
45 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
28 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
36 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
53 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
30 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
43 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
51 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
36 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
112 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 ...