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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Construction of a quadratic form under inequality constraints

I'm interested in the following question : Let $A \in \mathcal{S}_d(\mathbb{R})^{++}$ be a positive definite symmetric matrix. Let $z \in \mathbb{R}^d$ with $z \neq 0$ and $f \in \mathbb{R}^d$. Can ...
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proof that the bilinear form $a(u,v)$ is coercive

I am trying to show that $ a(u,v) = \int_{-1}^{1} \omega u_x v_x dx $ is a coercive and bounded bilinear form to apply the Lax Milgram Theorem. $\omega$ is defined as $\omega(x) := \sqrt{1-x^2}$. I ...
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constrained minimization problem for mixed finite element method

I'm trying to prove that a mixed finite element formulation's solution is equivalent to a constrained minimization problem's solution. Let $V$ and $Q$ be Hilbert spaces, and let $a : V \times V \to ...
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If the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?

I have a question: if the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?
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Equivalence of Lyapunov equation for continuous and discrete case

I am currently studying the original discrete/continous equivalence proof of the Lyapunov equation by Rice 1967. Continuous case: $A^\star L + L A = -C$ for $C \succcurlyeq 0$ Discrete case: $A^\...
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Set of isotropic vectors of generalized inner product space V is linear iff V over a field with char 2?

So, I've been learning linear algebra from a rather old book I really like (Andrzej Białynicki-Birula, "Geometria z algebrą liniową") that talks a bit about "generalized inner products" (that is, ...
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how can I show that a Function is Bilinear?

I have to Show, if a function $f(u,v)=1+3x_{1}-2y_{2}$ is Bilinear. $u^T =(x_{1},x_{2}) v^T =(y_{1},y_{2}) $ I know, that I have to prove $ ⟨ v , w 1 + w 2 ⟩ = ⟨ v , w 1 ⟩ + ⟨ v , w 2 ⟩ \\ {\...
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determing the possible signature of a bilinear form

I have the following question with me: "Let B be a nondegenerate symmetric form on a real vector space V of dimension d. Suppose max {dim U : U is an isotropic subspace of V } = m Determine the ...
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Proof of a sufficient condition of symmetric linear function.

If $f$ is a bilinear function and $\forall \alpha,\beta$ satisty$f(\alpha,\beta)=0 \Leftrightarrow f(\beta,\alpha)=0$, how to prove that $f$ is a symmetric or antisymmetric bilinear function? This ...
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How to find matrix of antisymmetrization $\pi_A(g)$ where $g$ is the bilinear form $e^1\otimes e^1-e^1\otimes e^2+3e^2\otimes e^1+2e^2\otimes e^2$

Basis $M=\{(3,1),(2,1)\}$. I solved that the dual basis \begin{equation} M^*=\{e^1,e^2\}=\{(1,-2),(-1,3)\} \end{equation} Then I solved that matrix \begin{equation} g=\left(\begin{array}{cc}1&-...
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Show that $\dim U + \dim U^\bot \geq \dim V$.

$V$ is a finite dimensional vector space over $F$. Given a symmetric bilinear form $b$ with $U$ a subspace of $V$ show $$\dim U + \dim U^\bot \geq \dim V.$$ Here I cannot assume that $b$ is non-...
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left and right radicals of a bilinear map

I have the following question with me: Let $V$, $W$ be finite dimensional vector spaces over F. For a bilinear map $B$ : $V × W$ → $F$, define its left and right radicals by: $Lrad(B)$ = {$v ∈ V : B(...
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Show that in general $\inf \sup \ne \sup \inf$ for bilinear functions

I am working on the following exercise: Let $K \subseteq \mathbb{R}^n$, $L \subseteq \mathbb{R}^m$ and let $F(K,L) \rightarrow \mathbb{R}$ with $$ F(x, \lambda) := c^Tx + \lambda^Tb - \lambda ...
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Is every bilinear form a linear transformation?

Let $B: \mathbb{R}×\mathbb{R}\to\mathbb{R}$ be a map defined by $B(a,b) = ab.$ This map $B$ is linear in both the coordinates but it is not linear as: $B((a,b)+(x,y)) = B(a+x,b+y) = (a+x)(b+y) \neq B(...
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dimension of symplectic lie algebra

The symplectic Lie algebra is defined as follows. Let dim($V$)$=2n$. We define a skew symmetrical bilinear form $f$ on $V$ with the matrix $S:=$ $\left( \begin{array}{rrrr} 0 & I_n \\ -I_n &...
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Conceitual difference between $\delta_{ij}$ and $\delta^i_j$

Question How is related $\delta_{ij}$ with $\delta^i_j$ ? Here $\delta_{ij}= \begin{cases} 1 \qquad\text{if} \qquad i=j \\0 \qquad \text{if} \qquad i\neq j\end{cases}$ Context I'm watching this ...
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Prove or disprove: $\phi(Ax,y)=\phi(x,A^Ty)$

Let $\phi$ a bilinear form. Then $\phi(Ax,y)=\phi(x,A^Ty)$ for each $A \in M_n(\mathbb{K})$ I assumed that this statement is true (I couldn't find counterexamples). So I tried proving this by the ...
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Negative definite bilinear form

Let $n$ be a positive integer, and let $V$ be the space of all $n \times n$ matrices over the field of complex numbers. Define a bilinear form $f$ on $V$ by $$f(A, B) = n \, \mbox{tr} (AB) - \mbox{tr}(...
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Total derivative of $\psi(x) = \frac{1}{\left \| x \right \|^p}Ax$

I have been trying to find the Fréchet derivative of the following function: $\psi(x) = \frac{1}{\left \| x \right \|^p}Ax$ $(x \in \mathbb{R}^n, A \in\mathbb{R^{m \times n}})$. One possibility would ...
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Symplectic Forms on Abelian Groups

If $G$ is finite group such that $G/Z(G)$ is abelian (i.e., $G$ is nilpotent of class at most 2) then the commutator function $$\varphi\colon G/Z(G)\times G/Z(G)\to G^\prime$$ given by $\varphi(g,h)=g^...
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How to visualize symplectic transformations?

This is a follow up question to this question. Let $\omega$ be an skew-symmetric bilinear form on $\mathbb{R}^{2n}$, which is unique up to change of basis. It is given by the formula $$\omega(\mathbf{...
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Did I mess up the nested summation?

Problem: Let $H$ be a bilinear function on an $n$-dimensional real vector space $V$. Fix a basis $\{b_1, \ldots, b_n\}$ for $V$, and let the matrix representation of $H$ under this basis be $B$ ...
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What is the matrix derivative of a symmetric bilinear form $\mathbf{a}^T X \mathbf{b}$ wrt $X$?

What is the derivative of the symmetric bilinear form $$ f_X(\mathbf{a},\mathbf{b}) = \mathbf{a}^T X \mathbf{b} $$ with respect to the (symmetric) matrix $X$? Following Wikipedia, and using the ...
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Orthonormal Basis of $\mathbb{R}^4$

I'm working on a problem that states: Find an orthonormal Basis of $\mathbb{R}^4$ with respect to the bilinear form defined through $$P= \begin{pmatrix}1&-2&1&-1\\-2&13&-6&...
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How do I define a pairing $\langle\cdot,\cdot\rangle$ on $\textbf{R}^{3}$?

Define a pairing $\langle\cdot,\cdot\rangle$, based on the given matrix $A$, on $\textbf{R}^{3}\times\textbf{R}^{3}$ by $\langle v,w\rangle = v^{T}Aw$, where \begin{align*} A = \begin{bmatrix} 1 & ...
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Symmetric bilinear form over $Mat_{2x2}(\mathbb{R})$

Let $V = Mat_{2x2}(\mathbb{R})$. Define $\varphi(A,B) := det(A+B) - det(A) - det(B)$. I have to show that $\varphi(A,B)$ is a symmetric bilinear form. It is easy to see that $\varphi$ is symmetric. ...
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Scalar Product on $\mathbb{R}^3$

I have a question on how to solve the following problem. Find a scalar product on $V=\mathbb{R}^3$ such that $\{e_1, e_2, e_3\}$ defined as $e_1=(1,1,0)$, $e_2=(1,0,1)$, $e_3=(0,1,1)$ is an ...
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Symmetric bilinear form, Compute V orthogonal

Given Task I‘m struggling with this exercise here. I know: $$\phi(v,w) = v^tAw$$ is a symmetric bilinear form V = span((1,0,0,0),(1,1,1,0)) $$V^t$$ means V orthogonal, so all the vectors s.t. = 0. ...
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Timoshenko beam bilinear form non-negative

The Timoshenko beam theory yields the following variational problem. For $t\in(0,1), f\in L_2$ given, we search for a $(w,\beta)\in V\subset[H^1(0,1)]^2$ such that it is a solution of \begin{align*...
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Canonical form of a Quadratic form.

I've been given the following quadratic form to find the canonical form of: $$ Q(\bf{z})= z_1z_2 + 2z_2z_3 − 3z_3z_4 $$ through the method of forming perfect squares. The method I've been taugh/show ...
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Eigenvalues of a symmetric bilinear form can be obtained as maximum and minimum values taken by the form on the circle of unit vectors

This is a claim in the book I am reading on Differential Geometry (Curves and Surfaces, 2nd Edition, by Montiel and Ros): The eigenvalues of a symmetric bilinear form on a two-dimensional Euclidean ...
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Real Euclidean space axiom violation

Assume that we have 4 usual axioms for the inner product (denoted as $(x, y)$ for elements $x, y \in V$ and some scalar $c$) in the real Euclidean space $V$: $(x, y) = (y, x)$ $(x, y + z) = (x, y) +...
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If $T$ is a symmetric bilinear form on vector space $V$, and let $U$ be a finite dimensional subspace of $V$, then $V=U+U^{\bot}$

Here is the full question: If $T$ is a symmetric bilinear form on vector space $V$, and let $U$ be a finite dimensional subspace of $V$, then $V=U+U^{\bot}$, where $U^{\bot}$ is the orthogonal ...
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Approximation of inf-sup stable variational problems

Consider flowing exercise: Let V be Hilbert space and $A: V\times V \rightarrow \mathbb{R}$ be a symmetric, elliptic (with constant $\alpha_1$) and continuous (with constant $\alpha_2$) bilinear ...
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Proof that bilinear form is coercive

Hi everyone I'm stuck with a proof and would be happy if anyone could help me out. Let V be a Hilbert Space and $A:V\times V \rightarrow \mathbb{R}$ symmetric, elliptic (coercive with constant $\...
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Injectivity restriction bilinear form

Let $V$ be a real vector space of dimension $n$ and $a:V\times V\to\mathbb R$ be a bilinear form. We define the associated matrix $A\in\mathbb R^{n\times n}$ as follows: $$ \langle A x,y \rangle = a(...
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Find the values of parameter a so that matrices A and B

Find the values of parameter a so that matrices $A=\begin{pmatrix} 1 &4-a-a^2 \\ 2 & -1 \end{pmatrix}$ and $B=\begin{pmatrix} -a-1 &3 \\ 3 & -5 \end{pmatrix}$ may represent the ...
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Quadratic Space Split over $\mathbb{R}$ implies Split over $\mathbb{Z}$?

Let $W \simeq \mathbb{Z}^{2n+1}$ be a free abelian group of rank $2n+1$ equipped with a symmetric nondegenerate bilinear form $\langle -, - \rangle \colon W \times W \to \mathbb{Z}$ of unit ...
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Vectorizing blocks of bilinear form

Math genius here. I have this problem of vectorizing the block matrix of bilinear entries as follows: $$W=\begin{bmatrix} e_1M_{11}e_1^T & e_1M_{12}e_2^T\\e_2M_{21}e_1^T&e_2M_{22}e_2^T \end{...
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Does it make sense to define a structure that is an inner product space and a K-algebra? And would this be useful in any way?

I've recently encountered K-algebras which I understand are different from inner product spaces in that one is equipped with a billinear product and the other with a bilinear form. Is it possible and/...
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Why into Minkowski metric we use a pseudo-Riemannian metric and not, simply, a pseudo-metric?

A pseudo-Riemannian “metric” is a nondegenerate quadratic form on a real vector space $R^n$ A pseudometric space $(X,d)$ is a set $X$ together with a non-negative real-valued function $d\colon X \...
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Condition when $V=W+W^{\perp}$ for $\dim V<\infty$

Let $\mathbb{k}$ be a field. Let $V$ be a finite-dimensional $\mathbb{k}$-vector space. Let $\varphi:V\times V\to \mathbb{k}$ ($\operatorname{char}\mathbb{k}\neq 2$) is bilinear (symmetric or skew-...
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Prove boundedness for this bilinear form

I have the following bilinear form $$a(u,v) = \int_0^1 \bigg(\frac{du}{dx}\frac{dv}{dx}+v\frac{du}{dx}+uv \bigg) \, dx$$ defined on $H^1_0(0,1)$ where $u(0)=v(0)=u(1)=v(1)=0$. How do I show that ...
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Notations and Definitions clarification in the Algebraic Duality of $c_0$ and $\ell_1$

I have a total of $3$ questions, my purpose is to clear up notation in the following question, since they are not introduced in the text and I cannot seem to find the notations online Below, we ...
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Tensor inner product vs Vector space inner product

How does the definition of inner product presented in this article: https://en.wikipedia.org/wiki/Inner_product_space relate to the definition presented in this video: https://www.youtube.com/watch?...
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86 views

Weak form of heat equation with Neumann boundary conditions

I'm trying to wrap my head around the ways to approach the heat problem. This time, with Neumann boundary conditions. Now, I have the theory down, but the only example I found leaves me behind by ...
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Bilinear forms on finite-dimensional space

Reading Quadratic functions in geometry, topology, and M-theory by M. J. Hopkins and I. Singer (https://projecteuclid.org/download/pdf_1/euclid.jdg/1143642908), I reckon some unclarity on p. 410: ...
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39 views

Bilinear form for general linear group

I have the statement in the book " 3-d transposition groups" by Michael Aschbacher. What does this mean? Does this mean that the general linear group can be regarded as the isometry group of a null ...
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Finding the dimension of the eigenspace of a bilinear form

Given a symmetric, bilinear form $\phi: V \times V \rightarrow \mathbb{R}$ with $V$ having dimension $n$. The map is given as $f_b(x)=x+\phi(x,b).b$ where $b \in V$, $b \neq 0$, $f_b: V \rightarrow ...
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Group of isometries isomorphic to special linear group

Given a 2-dimensional vector space $V$ over a field $\mathbb{F}$ with characteristic different from 2. I need to prove that the group of isometries of $f$, $Sp(V,f)$, is isomorphic to the special ...

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