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

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
33 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
0answers
20 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 ...
0
votes
2answers
26 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
20 views

Generators of Special Linear Matrix ???

This is a simple question, anyone can help: Can one generate this matrix $A_1$ or $A_2$ or $A_3$ from two matrices $B$, $C$ and their inverse ($B^{-1}$, $C^{-1}$): $$ A_1=\begin{pmatrix} 0& ...
0
votes
0answers
24 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
33 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
45 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
11 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 $\pm 1$?
4
votes
0answers
62 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
30 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
48 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
20 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
53 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
29 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? ...
0
votes
1answer
39 views

Definition of “symmetric bilinear (real) form indefinite”

In my studies I use these definition: Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form positive definite if 1) $\forall x \in e(f(x,x)\geq0)$ 2) ...
0
votes
1answer
49 views

Bilinear form on vector space

Does there exists a vector space $V$ and a bilinear form $w$ on $V\oplus V$ such that $w$ is not identically zero but $w (x,x) =0$ for every $x \in V$? My work is : if $M_2$ spanned by $\{(1,0,0,0), ...
1
vote
0answers
15 views

Checking degenerate and non degenerate

in P_n, w(x,y) = x(1) y(1) How to check whether it is degenerate or non-degenerate? I know how to show it is a bilinear form. If it is non-degenerate, I give out a case in which w(x,y) !=0 . Am not ...
2
votes
0answers
64 views

Invariant bilinear forms on Lie algebras

Consider a (compact) Lie group $H$ that acts on its Lie algebra $\mathfrak h$ in the usual way, $x\mapsto gxg^{-1}$ for any $x\in\mathfrak h$ and $g\in H$. Suppose we are given a real symmetric ...
0
votes
1answer
36 views

Existence of dual basis for an algebra with a nondegenerate bilinear pairing

Let $A$ be an algebra over a field $k$. Suppose that we have a non degenerate bilinear pairing $\beta:A \otimes A \to k$. Let $\{a_i\}$ be a basis of $A$. I would like to show that there exist a ...
-1
votes
1answer
38 views

Signature of quadratic form and eigenvalues

I'm asking about the signature of the quadratic form - the triple (n0, n+, n−). Is it true that n+ is the number of positive eigenvalues, and n- is the number of negative of eigenvalues of the matrix ...
2
votes
1answer
56 views

Quadratic form and symmetric bi linear form formula, basic point unclear to me

Something really basic but I have to ask it: I was taught that the formula of symmetric bi linear form of the quadratic form if: $f(v,w) = 1/2(q(v+w)-q(v)-q(w))$ but $q$ is linear so what did we get ...
0
votes
1answer
29 views

Bilinear form dimension

I saw that the dimension of a Bilinear form is: $$ \dim B(V,W) = \dim V \cdot \dim W $$ where a bilinear form means (in my course): $$ B : V\times W \rightarrow F $$ I don't really understand why, ...
0
votes
1answer
25 views

Symmetric bilinear and determinant

let $f: \Bbb R^2 \times \Bbb R^2 \to \Bbb R$ a symmetric bilinear form. s.t $\forall g\in M_2 (\Bbb R) \forall u,v\in \Bbb R^2 : f(gu,gv)=f(u,v)$ Find the possible values of $\det (g) $. Thoughts I ...
0
votes
1answer
51 views

Coercivity of bilinear form

I want to show that there is a unique solution for $$-u''=f$$ with boundary condition $$-u'(0)+u(0)=u'(1)=0$$ so I define bilinear form $$a(v,w) = \int\limits_0^1 {v'} w'dx + v(0)w(0)$$ so I should ...
2
votes
1answer
48 views

Bilinear form and weak sector condition

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
3
votes
1answer
208 views

Bilinear form with symmetric “perpendicular” relation is either symmetric or skew-symmetric

Let $b$ be a bilinear form on a finite-dimension vector space $V$ (over a field with char $\neq$ 2) such that for each $x,y\in V$ one has $b(x,y)=0\Leftrightarrow b(y,x)=0$. Prove that $b$ is ...
0
votes
2answers
54 views

Congruent diagonal matrix

For two days I reflect on this question without an answer: If $A=(i+j-1)_{1\le i,j\le n}$ is matrix in $\mathcal M_n(\mathbb R)$, the question is to find basis in which $A$ is congruent to diagonal ...
1
vote
1answer
34 views

Signature and bilinear forms

Suppose that given is: $A_{x}=\begin{bmatrix} x & -1 \\ -1 & x\\ \end{bmatrix}$ What is the signature of $A_1$ and $A_2$? Is it simple as I think: (1,0,1) and (0,1,1), because the first one ...
0
votes
2answers
38 views

let $B(,)$ and $B'(,)$ be inner products over $\mathbb R$. show there is $c \in \mathbb R$ s.t $B(u,u) \leq cB'(u,u)$

As the title says, let $B(,)$ and $B'(,)$ be inner products over $\mathbb R^n$. show there is $c \in \mathbb R^n$ s.t $B(u,u) \leq cB'(u,u)$ for all $u \in \mathbb R^n$. I have been thinking about ...
1
vote
2answers
65 views

Is it true that the whole space is the direct sum of a subspace and its orthogonal space?

Problem The ground field is $K$, $\operatorname{char}K\neq2$. Suppose $W$ is a (maybe infinite dimensional) subspace of a vector space $V$ with a symmetric/symplectic form ...
0
votes
1answer
39 views

Finding diagonal transformation matrix of a bilinear form

Let $f:\mathbb R^3 \times \mathbb R^3 \rightarrow \mathbb R$ be a symmetric bilinear form, and let $q$ be its quadric form, so that $q(x, y, z)= xy+yz$. Find the transformation matrix $A$ of $f$ by ...
0
votes
1answer
20 views

Question about definite positive symmetric bilinear form

Let $A$ be the matrix of a positive definite symmetric bilinear form. Prove $a_{11}a_{nn}\ge a_{1n}a_{n1}$. I don't really have a clue of how to solve this.
1
vote
0answers
109 views

Cauchy Schwarz inequality for random vectors

If $X$ and $Y$ are random scalars, then Cauchy-Schwarz says that $$| \mathrm{Cov}(X,Y) | \le \mathrm{Var}(X)^{1/2}\mathrm{Var}(Y)^{1/2}.$$ If $X$ , $Y \in \mathrm{R}^n$ are random vectors, is there a ...
1
vote
1answer
31 views

Associated Bilinear Form to Q (Quadratic Form)

I need to diagonalize the quadratic form $Q(x) = {x_{1}}^{2} + 2x_{1}x_{2} + 2{x_{2}}^{2} + 2x_{2}x_{3} + {x_{3}}^{2}$ so I know I need to find the associated Bilinear form with $B(x,x) = Q(x)$ - the ...
3
votes
2answers
54 views

Is the following bilinear form positive

Let $B:V\times V\rightarrow \mathbb R$ where $V$ is all the polynomials with degree $\leq2$. Determine whether the following is positive $\langle f,g\rangle =\int^1_0f(x)g(x)xdx$. I tried refuting ...
0
votes
2answers
41 views

How I can find the matrix $A$ of this quadratic form?

Let $(e_1,\ldots,e_n)$ the standard basis of $\mathbb R^n$ and we consider the quadratic form $$\Phi(x)=\sum_{1\le i<j\le n}(x_i-x_j)^2$$ How I can find the matrix $A$ of this quadratic form? My ...
4
votes
0answers
103 views

Non-degenerate bilinear forms of Lie algebra with a degenerate Killing form

Definition: A Lie algebra is defined by: $$ [e_a,e_b]={f_{ab}}^ce_c $$ The Killing form is $$ g_{ab}=-{f_{ac}}^d {f_{bd}}^c $$ Set-Up: The type of Lie algebra of our interests (found out during a ...
2
votes
0answers
19 views

Maximal noncompact forms in classical Lie algebra?

In this short note on Lie algebra, discussing about classical Lie algebra A,B,C,D class, in page 4 after Eq.(7), on the part of B,D class of O(2n,F) and O(2n+1,F) group (or algebra?), there is a ...
2
votes
1answer
82 views

wanted: a lucid demonstration that these $m+n$ bilinear equations have a solution

I shall first state the problem, and if it arouses any interest, I may gradually add a few notes, to provide a little context for those who, like myself, appreciate such background information. let me ...
4
votes
1answer
102 views

Matrix of a bilinear form <A,B> = tr(AB)

In revision for an upcoming exam, I've come across the following question: Let the bilinear form (A,B) be defined as tr(AB) on the space of 2x2 real matrices. Find an orthogonal basis for the form. I ...
1
vote
0answers
36 views

Estimate difference of solutions of an equation with a bilinear symmetric continuous form

My question refers to differential equations in Sobolev spaces. It is as follows: Let $\Omega \subset \mathbb{R}^n$ be a bounded open set. Let $a: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow ...
0
votes
0answers
17 views

Proof using Lax-Milgram with Backward Eulers method

I've got the next variational formulation of a problem: $(\delta u_i, \phi) + (\nabla u_i, \nabla \phi) + (u_i,1)(\phi,1) = (f(u_{i-1}), \phi). $ We want that prove that $u_i$ exists and in unique. ...
0
votes
0answers
42 views

Nondegenerate bilinear map

Let $V$ be the vector space consists of all $n\times n$ real matrices, and $f$ is a nonvanishing linear function on $V$ such that $$f(AB)=f(BA),\ \forall\ A,B\in V.$$ Show that $g(A,B)=f(AB)$ is a ...
3
votes
1answer
83 views

Show that a Bilinear form is Coercive

I'm reading through Brezis' book on functional analysis, Sobolev spaces and PDE, and I'm having trouble showing that the Bilinear form: $a(u,v) = \int_{0}^{2}u'v'dx+\left(\int_{0}^{1}u dx\right) ...
4
votes
2answers
58 views

Getting a positive semi-definite matrix from two positive definite matrices

Suppose I have two positive-definite Hermitian matrices $A$ and $B$. Their eigenvalues are strictly positive reals. Consider the matrices $A-tB$ for $0 \le t < \infty$. My goal is to conclude that ...
1
vote
0answers
14 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
2
votes
1answer
41 views

non degenerate bilinear map for modules

Let $R$ be a commutative ring with 1. Suppose $A,B$ are $R$-modules, $P:A\times B\to R$ is a bilinear map that satisfies the following property: if $P(a,b)=0$ for all $b\in B$, then $a=0$. Then is the ...
0
votes
0answers
70 views

Inner product and canonical forms

If $V$ is a vector space with finite dimension and for some symmetric bilinear form $f: V \times V \rightarrow \mathbb{R}$, how to I show that $f$ defines an inner product iff the unique real ...