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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Describe the space of solutions to a simple matrix equation

I would like to describe the space of solutions to the following matrix equation. Here $U$ and $V$ are two unknown real matrices with $n$ lines and $p$ columns (i.e. they belong to ...
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Matrix representation of a bilinear form

Let $a:\,\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a bilinear operator (i.e. $a$ is a linear operator in each component). Then exists a square matrix $A\in \mathbb{R}^{n\times n}$ such ...
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Uniqueness of positive-definite subspace

Let $g$ be a bilinear form over $\mathbb{R}^8$ with signature $(5,3)$. True or false - There exists a unique five dimensional subspace such that $g$ is positive-definite on this subspace. My ...
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Linear independence of isotropic vectors relative to a bilinear form.

Let $V$ be a vector space over $\mathbb{F}$, and let $g: V\times V\to \mathbb{F}$ be a bilinear form. A vector $v\in V$ is isotropic if $g(v,v) = 0$. Let $v,w\in V$ be 2 isotropic vectors such that ...
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differentiability of a function consisting of a bilinear form

Let $A \in M_{n, n}(\mathbb{R})$ be a symmetric matrix, and let $N = \{x \in \mathbb{R}^n \mid x^t A x = 0\}$. I first want to show that $N$ is closed. Next, I want to find an explanation why $f(x) = ...
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Bilinear forms have same rank

Exercise 12, Chapter 10, sec 10.4 [Hoffman and Kunze]. Let $V$ be a finite-dimentional vector space over a subfield of the complex numbers, let f,g be skew-symmetric bilinear forms on V.Show that ...
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15 views

Non-degenerated restricted bilinear form

Studying bilinear forms, this question came to my mind: If $f$ is a non-degenerated bilinear form in $V$ and $W$ is a subspace of $V$, is $f$ restricted to $W$ is also non-degenerated? I know that ...
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Showing that a bilinear form is non degenerate

Given a finite dimensional vector space $V$ over $F$ and a fixed matrix $(\alpha_{ij})=A \in M_n(F)$ and the bilinear form on $V \times V$ by $B(u,v)=\sum_{i=1}^{n} \sum_{j=1}^{n} \alpha_{ij} ...
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Linear transformation and bilinear forms

I am trying to do the next exercise (Exercise 11, section 10.2, Hoffman & Kunze). I defined a linear functional $L: V \to V^*$ as $(L_f\alpha)(\beta)=f(\alpha, \beta)$. It is easy to show that ...
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Generalized criterion for positive definiteness of real symmetric matrices [duplicate]

I searched for the following question on stack exchange but couldnot find the general case answered or asked anywhere. This is an exercise from Artin's book, under the chapter named "Bilinear forms''. ...
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Bilinear forms on vector spaces over field extension

Could anyone please help? Let K be an algebraic number field, and consider the symplectic form on $B: K^2 \times K^2 \to \mathbb{Q}$ given by $B((k_1,k_2),(l_1,l_2)) = Tr_{K/ \mathbb{Q}}(k_1 l_2 - ...
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On bilinear forms: if $B$ is non-degenerate, then $(U_1\cap U_2)^{\perp_L} = U_1^{\perp_L} + U_2^{\perp_L}$

I think I have a proof, but I'm not completely certain it is correct: Because $B$ is non-degenerate, it follows that if $W \leq V$, then $W = W^{\perp_L \perp_R}$. It is evident, too, that $(U_1 + ...
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Complement of the radical of a bilinear form

The following is given(V is a vector space and $\gamma$ is a bilinear form: $rad(V)=\{v\in V|\;\gamma(v,w)=0 $ for all $ w \in W\}$ Let U be the complement of rad(V), show that ...
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determining a linear isomorphism so that two quadratic forms become equivalent

Consider the matrix $$ G = \begin{pmatrix} 3 & 1 & -2 \\ 1 & 2 & 0 \\ -2 & 0 & -3 \\ \end{pmatrix}$$ and the quadratic form $q: \mathbb{R}^3 \to \mathbb{R}$, given by $q(v) ...
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Dual Space of an Euclidian Space is also Euclidic with a specific bilinear form

Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form, and let $\overline{\gamma}$ be defined by: $$\overline{\gamma}: V^* \times V^* \to K, \gamma(x, y) = ...
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Properties of bilinear forms

Let $V$ be a real vector space with norm $\|\cdot\|_V$ and $W$ a closed, linear subspace of $V$. A bilinear form $a\colon V\times V\rightarrow \mathbb{R}$ is called symmetric, if $a(u, v) = ...
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Show that a field extension $L/K$ is separable iff the trace form is non-degenerate.

Let $L$ be a finite field extension of $K$. I have the following question: Show that $L/K$ is separable if and only if the bilinear trace form $\text{Tr}_{L/K}:L\times L \to K$ is non-degenerate. ...
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isometric quadratic spaces over a prime field

Let $(V, \gamma)$ be a quadratic space, where $V$ is an $n$-dimensional $\mathbb{Z}/(7)$-vector space and $r = r(\gamma)$ is the rank of the bilinear form. I want to show: either, $(V, \gamma)$ is ...
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Each bilinear form induces a unique bilinear form from the dual space

Let $V$ be a finite dimensional vector space over a field $K$. Let $\gamma: V \times V \to K$ be a nondegenerate bilinear form. I now want to show that there exists one and only one bilinear form ...
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A Theorem on Skew-Symmetric Bilinear Forms in Hoffman and Kunze's Book

On pg. 377 in Hoffman and Kunze's Linear Algebra(Second Edition) Theorem 6 reads: Let $V$ be an $n$-dimensional vector space over a subfield of the field of complex numbers, and $f$ be a skew ...
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If a Bilinear Form is Non-Degenerate on a Subspace $W$, then $V=W\oplus W^\perp$.

$\newcommand{\range}{\text{image}}\newcommand{\ann}{\text{Ann}}\newcommand{\set}[1]{\{#1\}}$ Problem: Let $V$ be a finite dimensional vector space over a field $F$ and $f$ be a symmetric bilinear ...
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a bilinear form is always the sum of two others

Let $K$ be a field with a characteristic, other than 2. Let $V$ be a finite dimensional vector space over $K$, and let $\gamma: V \times V \to K$ be a bilinear form. I now want to show that there ...
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Surjectivity of an alternating, non degenerate bilineair form

I have a function $\varphi: V \times V \to k$ where $V$ is a vector space over ground field $k$ (although I would also like to prove this for $R$-modules). I know that $\varphi$ is alternating, that ...
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Find all complex matrices $A$ such that $n\operatorname{Tr}(AB) = \operatorname{Tr}(A)\operatorname{Tr}(B)$ for all $B$. [duplicate]

Consider a bilinear form $f(A,B) = n\operatorname{Tr}(AB) - \operatorname{Tr}(A)\operatorname{Tr}(B)$ defined on $M_n(\mathbb{C})$. I need to find the set $U^\perp$ of all matrices $A$ such that ...
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bilinear forms on $M_{n, n}(K)$

Let $K$ be a field and $V = M_{n, n}(K)$ the ring of $n \times n$ matrices over $K$. For any $f \in V^*$ (the dual space of $V$), we set: $\gamma_f: V \times V \to K, (A, B) \mapsto f(A B^t)$. I now ...
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Show that a bilinear form is complex skew-Hermitian.

A complex skew-Hermitian form (also called an antisymmetric sesquilinear form), is a complex sesquilinear form $s : V × V → \mathbb C$ such that $$s(w,z) = -\overline{s(z, w)}.$$ Prove that the ...
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Matrix Representation of Bilinear Form/Map in Matrix Space

I am trying to understand bilinear forms and have a related problem, but unfortunately all of the examples I've been able to find deal with $V = \mathbb{R}^n$. I am dealing with $V = M_{n \times ...
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Necessary and sufficient condition for a bilinear form to be symmetric

Given the bilinear form $f(A,B)=\operatorname{tr} (A^t M B)$ where $A,B$ are two $n\times n$ matrices I have to find a necessary and sufficient condition (on $M$) for $f$ to be symmetric. I found out ...
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Kernel of bilinear form

It is written in book, I read: kernel of bilinear form is space consisting of vectors $y$, such: $$Ker(\alpha)=\{y\in V:\alpha(x,y)=0,\ \forall x\in V\}$$ Nice I get it, but then it is said, that ...
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Bilinear forms and operators

I am introducing myself to bilinear forms. I would like a little help to do this exercise: Let $f$ and $g$ two bilinear forms on $V$, with $f$ non-degenerated. Show that there is an unique operator ...
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Is convolution a coercive bilinear form in $L^2$ -space?

This is one of the problems in functional analysis course I'm having. Suppose $f,g \in L^2(0,10)$. Then define a bilinear form $$ B:(f,g)\mapsto \int_0^{10} f(x)g(10-x) dx. $$ Now I have to find out ...
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Is there a convergence proof for ADMM applied to biconvex/bilinear problems?

I wonder if there is a local convergence proof for ADMM applied to biconvex problems? More specifically, my problem is as follows: $\text{minimize}_{x,y} f(x) + g(y) + \| y \circ Ax \|_2^2 $ , ...
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On non-degenerate bilinear forms on infinite dimensional vector spaces

For any non-degenerate bilinear form $(\cdot,\cdot)$ on a vector space $V$ and a linear functional $f$, there exists $v \in V$ such that $f(v)=(v,w)$ for all $w \in V$. It's easy in ...
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Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
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Find the possible signatures of the bilinear forms

Find the possible signatures of the following bilinear forms: The bilinear form $\phi:\mathbb R^n\times\mathbb R^n\to\mathbb R$ given by $\phi(x,y)=x^Tp(A)y$ where $p(t)=t^2+bt+c$ is a ...
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If $H$ is a bilinear form then for every $x$ there exists non-null $y$ with $H(x,y)=0$

Prove or disprove: Suppose $H$ is a bilinear form on a finite dimensional vector space $V$, with $\dim(V)>1$. Then for any $x\in V$ there always exists a non-zero $y\in V$ such that $H(x,y)=0$. ...
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41 views

Constructing a bilinear form on $\mathbb{R}^2$ that gives rise to a particular matrix

As the title says, I'm trying to create a bilinear form $B(\cdot, \cdot)$ on $\mathbb{R}^2$ with some particular constraints (which I do not know as yet) related to the Lorentzian space ...
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What method was used here to expand $\ln(z)$?

On Wikipedia's entry for bilinear transform, there is this formula: \begin{align} s &= \frac{1}{T} \ln(z) \\[6pt] &= \frac{2}{T} \left[\frac{z-1}{z+1} + \frac{1}{3} \left( \frac{z-1}{z+1} ...
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Euclidean norm of complex vectors

I am working on a proof: One has two vectors, $u,v \in \mathbb C^n$, such that $u \cdot v=0$ . I am trying to prove that $$|u + v|^2 = |u|^2 + |v|^2.$$ I am a little stuck on how to do $u + v$ ...
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What does $[L]=[I]^{-1}[II]$ mean?

I have a question about one of the equations in my notes. Matrix representations of Weingarton map, first fundamental form and second fundamental form satisfies $[L]=[I]^{-1}[II]$ According to ...
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Prove that the quadratic covariation is a bilinear form

If we take $X,Y,Z$ to be square integrable martingales starting at zero, we want to show that for any $\alpha\in\mathbb{R}$ we have $\langle X + Y , Z \rangle = \langle X,Z\rangle + \langle Y, Z ...
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Expected value of a bilinear form

I read many of the previous posts but I could not find my answer yet. Let $x \in \mathcal{C}(0,\sigma^2_x)$ and $y \in \mathcal{C}(\bar{y},\sigma^2_y)$ be two $N \times 1$ column vectors of i.i.d. ...
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Question about bilinear pairing.

Let $V$ and $W$ be two $k$-vector spaces of dimension $n$ and let $\circ :V \times W \to k$ be a $k$-bilinear pairing that is nonsingular. If $\{v_1,..,v_n \}$ is a basis for $V$, how can I see that ...
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Finding the symplectic matrix in Williamson's theorem

tl;dr: How do I construct the symplectic matrix in Williamson's theorem? I am interested in a constructive proof/version of Williamson's theorem in symplectic linear algebra. Maybe I'm just missing ...
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Question on bilinear forms and invertible matrix

Let $k$ be a field and $V$,$W$ be two $k$-vectorspaces of same dimension. Now if $V \times W \longrightarrow k$ is a bilinear pairing, how can I see that this bilinear pairing is nonsingular if and ...
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How does the inner product (-,-) with norm ||.|| relate to bilinear and linear forms in the FEM?

I am attempting to prove the Lax-Milgram lemma for the weak formulation of the finite element method. However I first need to prove continuity of the bilinear and linear forms ($a(u,v)$ and $l(v)$), ...
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Orthogonal operator on $V\oplus V^*$ preserving projection on $V$

Let $V$ be a real finite dimensional vector space. $V\oplus V^*$ has a natural symmetric bilinear form: $$\langle X+\xi,Y+\eta\rangle=\frac{1}{2}(\xi(Y)+\eta(X)) $$ If $B\in \wedge^2V^*$, and we ...
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Diagonalization of a symmetric matrix over algebraically closed field

Let $k$ be an algebraically closed field. Let $A$ be an $n \times n$ symmetric matrix with entries in $k$. Does it then follow that there exist eigenvectors of $A$ which form an orthonormal basis of ...
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20 views

Finite quadratic variation leads to finite covariation

I'm trying to prove that if two functions have finite quadratic variation then their covariation is finite. I've seen that $2|[X,Y]_{t}| \leq [X]_{t}+[Y]_{t}$ but I can't see how to get there. It ...
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Bilinear functions

I have some trouble with notation of bilinear functions. I will state the theorem i have trouble with: A function $$\;\Bbb R^m\times \Bbb R^n\to\Bbb R\;$$ is bilinear if and only if it can be written ...