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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Dot product in bilinear form (Euclidean space) [closed]

Find $a$, a real number such as $$ B((x,y),(x',y'))=xx'+2xy'+2x'y+ayy'$$ is a dot product.
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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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1answer
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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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31 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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2answers
53 views

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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1answer
49 views

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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45 views

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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1answer
27 views

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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1answer
30 views

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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2answers
19 views

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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1answer
25 views

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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1answer
25 views

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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2answers
35 views

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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1answer
15 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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26 views

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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1answer
24 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 $\gamma \in GL(V)$ ...
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1answer
59 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 ...
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1answer
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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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1answer
28 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 $T^*T$ ...
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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
47 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 ...
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1answer
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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 ...
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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 ...
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1answer
48 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 ...
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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 ...
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52 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 ...
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1answer
61 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 ...
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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 ...
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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 ...
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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 ...
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1answer
51 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: ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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: ...
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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 ...