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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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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27 views

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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50 views

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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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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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 ...
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Show non-degenerate form of subspaces

Let $(E, d)$ be nonzero bilinear space over $K$ and place conditions: $$ d(x,y) = d(y,x) \\ d(x,y) = - d(y,x) $$ for every $x,y \in E$. Show that if $E_1$ and $E_2$ are singular (degenerate?) bilinear ...
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associated matrix to an orthonormal basis

Let T be a symmetric bilinear form. Given an orthonormal basis for the vector space, is the associated matrix the identity matrix? Thanks in advance.
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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 $\gamma \in GL(V)$ ...
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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 ...
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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 ...
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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 ...
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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 $T^*T$ ...
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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 $BX=Y$.
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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\ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
48 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 ...