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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Checking/testing bilinearity of a function

I have already aware of the bilinear form of a function which is: <x,y> =x^tAy= ∑_(i,j)=〖a_(i,j)x_iy_i〗 Applying this concept to an example: α(x,y)=x^T* [2 -1, *y -1 1] Assuming x = [x1 y1], ...
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Bivectors and (2,0)-tensors

Given a vector space $ V $ of finite dimension $n$, you can consider his dual $ V^* $ i.e. the space of the linear functionals $ \omega:V\longrightarrow K $ (where $K$ in the field of the scalars), or,...
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Bilinearity or not of a tensor product

If I have a tensor field $g$ of type $(0,2)$, in local coordinates expressed as $g=g_{ij}dx^i\otimes dx^j$, changing coordinates ($x^i\rightarrow y^j$) I obtain the following: $$ g=g_{ij}(\frac{\...
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Artin Chapter 7 1.13

The author basically concludes that, if $\forall X, Y\in\mathbb{F}^n: X^TAY= X^TBY$ then $A= B.$ This doesn't seem immediate to me. It is not hard to prove this, but unless I am missing something, the ...
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If $X \mapsto X^tG$ is an isomorphism, then $G$ is an invertible matrix. (Bilinear forms)

Let $f : E \times F \to R$ be a bilinear form that is non-singular on the left, where $E, F$ are free $R$-modules of dimension $n$ (both of them). Then if $X, Y$ are column vectors for general ...
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Bilinear form respect to the representation $V/W$ of Lie algebra

I got a problem from the book An Introduction to Lie Groups and Lie Algebras written by Kirillov. The exercise 5.1 says $$\begin{array}{l} \text { (1) Let } V \text { be a representation of } \...
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A blinear form $H$ is non-degenerate if and only if $H(v,w) = 0 \ \forall v \in V \Leftrightarrow w = 0$

Question: Given a bilinear form $H: V \times V \rightarrow F$ on a finite dimensional V, show that $H$ is nondegenrate if and only if the only $w \in V$ s.t. $H(v,w) = 0 \quad \forall v \in V$ is $w = ...
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Discrepancy in definitions of nondegenerate bilinear and sesquilinear form

The definition of a nondegenerate bilinear form F(x,y) in finite dimensions is given here to be $ F(x,y)=0 \; \; \forall \; y \; \; \Rightarrow \; \; x=0 \qquad \text{and} \qquad F(x,y)=0 \; \; \...
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Constant such that opposite direction of triangle inequality holds

Given a symmetric, bounded, elliptic bilinear form $a(\cdot,\cdot)$ on a space $V=V_1 \oplus V_2$ I want to show that there is a constant $c$ such that $$ \forall v = v_1 + v_2 \in V_1 \oplus V_2: a(...
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Lie algebra (invariant)-action on bilinear form

Let $(\pi, V)$ be a rational representation of $G$ a Lie algebra, and let $B$ be a bilinear form defined on $V$. Then there is an action of $G$ on $B$ given by $$ g \cdot B(u,v) = B(g^{-1}u, g^{-1}v).$...
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Given nondegenerate symmetric bilinear form $f$, how is $L_f(x) = f(x,-)$ defined for $x = 0$?

Let $f$ be a nondegenerate symmetric bilinear form over an $n$-dimensional vector space $X$. For each nonzero vector $x \in X$, define the dual vector $$f(x,-) \in X^{\prime}, \hspace{3mm} f(x,-): y \...
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Bilinear forms and polarization identity

Let $H$ be a Hilbert space and $\sigma: H \times H \to \mathbb{C}$ a sesquilinear form (linear in the first variable, anti-linear in the second variable). Then it is well-known that the following ...
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Probability distributions on a pseudo Euclidean space

I would like to know if there are standard probability densities with explicit expressions, which are functions of a pseudo-scalar product. On $\mathbb{R}^2$, $$ f(x,y)=k(x^2-y^2)\quad \text{with} \...
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Reflection on Hyperplane

Let $(V, \langle,\rangle) $ be an euclidean Vector space. For $w \in V, w\ne 0$, the map $s_w: V \rightarrow V$, $s_w (v): = v -2 \tfrac{\langle v,w \rangle}{\langle w,w \rangle}$ is defined (the ...
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A basis such that the bilinear form $\phi$ is diagonal, but the operator $T$ is not

Let $Q(x,y,z) = 2(xy+xz+yz) - (x^2+y^2+z^2)$ be a quadratic form in $\mathbb R^3$ and let $\phi$ be its associated bilinear form. Let also $T:\mathbb R^3 \rightarrow \mathbb R^3$ be a linear operator ...
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What is the relation between a matrix as a linear function versus the same matrix as a bilinear function?

Given an $n \times n$ matrix $A$, we can define a linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^n$ by $T(x)=Ax$. We could also define a bilnear function $T: \mathbb{R}^n \times \mathbb{...
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Bilinear Form and Rank Proof

I found a task in the internet about bilinear forms that I would like to unterstand. I changed a few variables but it goes like this. Suppose that $\psi: U \times V \rightarrow K$ is a bilinear form ...
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Symmetric Bilinear Form/ Hermitian Form unique Matrix representation

Given $\mathbb{K} = \mathbb{R} $ or $ \mathbb{C}$ I have $\varphi:\mathbb{K^n}\times\mathbb{K^n}\rightarrow \mathbb{K}$ a symmetric Bilinear Form or complex Hermitian Form/Symmetric sesquilinear form ...
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How to define a map $V^* \to V$ arising from a non-degenerate Hermitian form on $V$?

I am studying Representation of finite groups from the book by Fulton and Harris. There I found the following statement in the form of lemma $3.35$ page no $:$ $40$ which I failed to understand ...
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Prove that $\dim W−\dim(W∩W^⊥)$ is an even number. [closed]

Suppose $f$ is a skew-symmetric bilinear function on a space $V$, $W$ is a subspace of $V$ and $W^⊥$ is its orthogonal complement with respect to $f$ Prove that $\dim W−\dim(W∩W^⊥)$ is an even number. ...
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Which of the following Maps is Bilinear?

Problem: Show which of the following maps are bilinear ? $\phi_1 : \left( \left(\begin{array}{c} v_1 \\ v_2 \end{array}\right) , \left(\begin{array}{c} w_1 \\w_2 \end{array}\right)\right) \mapsto ...
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What is the dimension of $w$?

Let, $V$ be the vector space of all $n \times n $ matrices over the field $\mathbb{C}$. We define a bilinear form as $f(A,B) = n \operatorname{tr}(AB)- \operatorname{tr}(A) \operatorname{tr}(B)$ Where ...
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37 views

When do we have orthogonal bases in a vector space equipped with a skew-symmetric form?

Suppose we have a vector space $V$ over some field $\mathbb F$ together with a skew-symmetric form $\langle\space ,\space \rangle$, i.e., $\langle\space ,\space \rangle$ is bilinear and $\langle v,v\...
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Why are some functions called 'forms'?

The question is simple: why are some functions called 'forms'? Modular 'forms', bilinear 'forms', differential 'forms', quadratic 'forms', and so forth. It is not concretely a mathematical question ...
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Frobenius-Schur indicator: why doesn't the Weyl unitary inner product qualify?

As best as I understand, the Frobenius Schur indicator tells us for a finite group $G$ and an irreducible representation $\rho: G \rightarrow \operatorname{Aut}(V)$ (where $V$ is a vector is a vector ...
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Bilinear forms on R^n: What is the exact meaning of the indices i and j for <ei,ej>

Let $f:\mathbf{R}^2\to\mathbf{R}^2$, $f\left(\begin{bmatrix}x_1 \\ y_1\end{bmatrix},\begin{bmatrix}x_2 \\ y_2\end{bmatrix}\right) = 2x_1x_2 + 3x_1y_2 + x_2y_1$. As an example we want to represent $f$ ...
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Connection between weak coercivity (Garding inequality) and inf-sup condition (Necas theorem)?

I am looking for a proof that weak coercivity (Garding inequality) and the condition $a(u,v) = 0, \forall v \in V \implies u = 0$ is equivalent (or implies) to the inf-sup condition and the condition $...
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Solve $A\mathbf{x}=\mathbf{v}(\mathbf{x}^TD\mathbf{v})$ for $A$

Is there a way to simplify $$f(\mathbf{x})=\mathbf{x}-\mathbf{v}(\mathbf{x}^TD\mathbf{v})$$ to the form $$f(\mathbf{x})=(I-A)\mathbf{x}$$ where $\mathbf{x}$ and $\mathbf{v}$ are column-vectors and $D$ ...
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Proof of Lemma 3.2.4 Cox, Little Schenck

I am reading the book Toric Varieties by Cox, Little and Schenck, and they stated the following Lemma without proof 'Let $\sigma$ a strongly convex rational polyhedral cone in $N_{\mathbb{R}}$. Let $...
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Pedantic question about symmetric/non-symmetric bilinear forms: is symmetry of the bilinear form redundant?

A quadratic form $q$ is defined in terms a symmetric bilinear form $f$ by $q(\mathbf{v}):=f(\mathbf{v},\mathbf{v}).$ However, this definition seems too restrictive. Why must $f$ be symmetric? Even if $...
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Maximum on circle through normal p.d.f in $\mathbb{R}^3$

Given a normal distribution on $\mathbb{R}^3$ $$p(\mathbf{x})\propto\exp(-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^T\Sigma^{-1}(\mathbf{x}-\mathbf{\mu}))$$ and a circular trajectory through $\mathbb{R}^3$ ...
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How can I show that $f$ is bilinear and onto?

Let $\{e_1,e_2,\cdots,e_n\}$ and $\{f_1,f_2, \cdots, f_m \}$ be the standard ordered bases for $\Bbb R^m$ and $\Bbb R^n$ respectively and let $\{E_{11},E_{12},\cdots,E_{mn} \}$ be the standard basis ...
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Can every isomorphism between a vector space and its dual be written as a non-degenerate bilinear form?

If I have understood correctly, non-degeneracy of a bilinear form: $$\omega:V\times V\rightarrow \Bbb F \tag{1},$$ in which $\Bbb F$ the underlying field of $V$, is a strong enough condition to ...
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Characterization of cross product (proof that only exists in dimension 3)

In order to show that the cross product exists only in $\mathbb R^3$, our teacher puts this remark, "Exercise left to the reader". I have no idea how to solve it. Here's the problem: "...
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Is there a name for “positive-semidefinite” quadratic forms when the base field is not $\mathbb R$?

Consider a vector space $V$, with a symmetric bilinear form $\cdot$ , over a field $\mathbb F$ with characteristic not $2$. Suppose, for any $x\in V$, if $x\cdot x=0$, then $x=0$. What is this ...
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23 views

coordinate tensor with respect to the standard basis vectors

Consider the bilinear form $\phi: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ defined by $$\phi(v,w)=\langle(1,0,0)^T,v \times w\rangle$$ Then what is the coordinate tensor with respect ...
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Given a positive definite bilinear form, show that there exists $\alpha > 0$ so that $a(x,x) \ge \alpha||x||_{2}^2$

Let $a : \mathbb{R}^n * \mathbb{R}^n \rightarrow \mathbb{R}$ be a bilinear form: $$a(x,y) = \sum_{i,j=1}^n a_{ij}x_{i}y_{j} \ \ \ \ \ \ \ a_{ij} \in \mathbb{R},\ \ x = \begin{pmatrix}x_1\\.\\.\\.\\x_n\...
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45 views

Bilinear forms on normed vector spaces

For a positive definite bilinear form a: $\mathbb{R}^n\times\mathbb{R}^n\to \mathbb{R}$ defined by: $$a(x,y)=\sum^n_{i,j=1}a_{ij}x_{i}y_{j}$$ $a_{ij}\in \mathbb{R}\ x,y,$ two vectors of length $n$, ...
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Chain differentiation of bilinear form.

Suppose one has a bilinear form $\alpha(x,y)$ and suppose one has the derivative $$\frac{\partial }{\partial t} \alpha(x,ty)$$ How does the chain differentiation work in this case? Is it $$\frac{\...
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If $f$ is a bilinear function then prove that $Df(a,b).(x,y)=f(a,y)+f(x,b)$

Let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^p$ be a bilinear function. Prove that $$Df(a,b).(x,y)=f(a,y)+f(x,b)$$ Here $Df(a,b)$ is the Jacobian. My Problem I'm not able to ...
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About possible signatures of a non-degenerate symmetric bilinear form.

Say, I have a non-degenerate symmetric bilinear form on a vector space with Dim = $4$. Suppose I have a set of basis $\{v_1,v_2,v_3,v_4\}$ and I know for any $i$, $<v_i,v_i>$ is positive. So ...
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Is there an analog of the spectral theorem for symmetric, non-degenerate bilinear forms?

I was thinking of persymmetric matrices after I saw this question. Persymmetric matrices are notable in that they satisfy $(Ax,y) = (x,Ay)$, where $(\cdot,\cdot)$ the bilinear form over $\Bbb R^{n}$ ...
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59 views

$tr(xy)$ is non-degenerate if and only if $A$ is absolutely semisimple

Suppose $A$ is any algebra over any groundfield $k$, and let $tr(u)$ denote the trace of the endomorphism of the underlying vector-space of $A$, $L_{u} : x \mapsto ux$. Then, I want to prove that : $...
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Why linear map from here from $V \to V^*$ is surjective.

Given a 2-covector $\omega$ over finite dimension vector space $V$, We assume the linear map $i(\omega):V\to V^* $,which is defined as $i(\omega)(v) = \omega(v,\star)\in V^*$,with $\omega$ is ...
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How to determian the linear independence of linear functionals?

Say, we have a vector space $V$ (assume dim $>=3$), over the field of rational numbers with the basis $v_1...v_n$. Now, if we define the linear functionals $g_i$ as $g_i(v_j) = j+(i-1)*(-1)^{ij}$, ...
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35 views

Find a base such that $f(x,y)=x_{1}y_1+2x_{1}y_{2}+2x_{2}y_{1}+4x_{2}y_2$ has a diagonal form

Let $f: \mathbb{R}^2\to \mathbb{R}$ be a bilinear form defined as: $$f\left((x_{1},x_{2}),(y_{1},y_{2})\right)=x_{1}y_{1}+2x_{1}y_{2}+2x_{2}y_{1}+4x_{2}y_{2}$$ Find a base for $\mathbb{R}^2$ such that ...
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Does the division of two continous variables become a bi-linear or non-linear term in Linear programming?

For example power factor PF constraint with the real power P and reactive power Q: tan(arc cos(-PF))<=Q/P<=tan(arc cos(PF)) PF is a fixed parameter say 0.8, PF range is {0,1} P and Q continuous ...
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36 views

Coercivity of a bilinear form and the associated differential equation

I have to prove that the following bilinear form is not coercive in $H_0^1(0,1)$. $$a(u,v)=\int_0^1x^2u'(x)v'(x)dx$$ Is it as simple as saying since there is no lower bound given on $x^2$ that is ...
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35 views

Index of a symmetric bilinear form on a real vector space

Let $V$ be a (not necessarily finite-dimensional) real vector space, and suppose $B:V\times V\to \Bbb R$ is a symmetric bilinear form on $V$. The index of $B$ is defined to be the maximum dimension of ...
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103 views

Find signature and the symmetrical bilinear forms of $\phi(x^2) $ & $\phi(x)^2$

I have the following problem I struggle with : Let $ \mathbb{K} $ a commutative field [of different characteristics of $2$ ( it means that $1+1=2$ has an inverse $\in \mathbb{K}$)]. A $\mathbb{K}\...

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