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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questions
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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 ...
1
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0answers
17 views
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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0answers
19 views
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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2answers
104 views
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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1answer
26 views
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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1answer
25 views
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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1answer
34 views
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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0answers
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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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0answers
16 views
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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1answer
37 views
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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0answers
23 views
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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votes
1answer
31 views
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(...
1
vote
1answer
20 views
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
$$ F(x, \lambda) := c^Tx + \lambda^Tb - \lambda ...
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1answer
32 views
Is every bilinear form a linear transformation?
Let $B: \mathbb{R}×\mathbb{R}\to\mathbb{R}$ be a map defined by $B(a,b) = ab.$
This map $B$ is linear in both the coordinates but it is not linear as:
$B((a,b)+(x,y)) = B(a+x,b+y) = (a+x)(b+y) \neq B(...
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vote
1answer
45 views
dimension of symplectic lie algebra
The symplectic Lie algebra is defined as follows.
Let dim($V$)$=2n$.
We define a skew symmetrical bilinear form $f$ on $V$ with the matrix $S:=$
$\left( \begin{array}{rrrr}
0 & I_n \\
-I_n &...
4
votes
1answer
37 views
Conceitual difference between $\delta_{ij}$ and $\delta^i_j$
Question
How is related $\delta_{ij}$ with $\delta^i_j$ ? Here $\delta_{ij}= \begin{cases} 1 \qquad\text{if} \qquad i=j \\0 \qquad \text{if} \qquad i\neq j\end{cases}$
Context
I'm watching this ...
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1answer
46 views
Prove or disprove: $\phi(Ax,y)=\phi(x,A^Ty)$
Let $\phi$ a bilinear form. Then $\phi(Ax,y)=\phi(x,A^Ty)$ for each $A \in M_n(\mathbb{K})$
I assumed that this statement is true (I couldn't find counterexamples). So I tried proving this by the ...
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0answers
27 views
Negative definite bilinear form
Let $n$ be a positive integer, and let $V$ be the space of all $n \times n$ matrices over the field of complex numbers. Define a bilinear form $f$ on $V$ by $$f(A, B) = n \, \mbox{tr} (AB) - \mbox{tr}(...
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1answer
45 views
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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0answers
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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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votes
2answers
75 views
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 $$\omega(\mathbf{...
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votes
0answers
24 views
Did I mess up the nested summation?
Problem:
Let $H$ be a bilinear function on an $n$-dimensional real vector space $V$. Fix a basis $\{b_1, \ldots, b_n\}$ for $V$, and let the matrix representation of $H$ under this basis be $B$ ...
5
votes
3answers
76 views
What is the matrix derivative of a symmetric bilinear form $\mathbf{a}^T X \mathbf{b}$ wrt $X$?
What is the derivative of the symmetric bilinear form
$$
f_X(\mathbf{a},\mathbf{b}) = \mathbf{a}^T X \mathbf{b}
$$
with respect to the (symmetric) matrix $X$?
Following Wikipedia, and using the ...
1
vote
2answers
85 views
Orthonormal Basis of $\mathbb{R}^4$
I'm working on a problem that states:
Find an orthonormal Basis of $\mathbb{R}^4$ with respect to the bilinear form defined through
$$P= \begin{pmatrix}1&-2&1&-1\\-2&13&-6&...
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1answer
26 views
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 & ...
1
vote
1answer
66 views
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. ...
0
votes
2answers
34 views
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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0answers
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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. ...
0
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1answer
31 views
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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votes
4answers
71 views
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 ...
1
vote
2answers
49 views
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 ...
2
votes
1answer
36 views
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) +...
1
vote
1answer
41 views
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 ...
0
votes
0answers
87 views
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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1answer
69 views
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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0answers
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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:
$$
\langle A x,y \rangle = a(...
1
vote
1answer
60 views
Find the values of parameter a so that matrices A and B
Find the values of parameter a so that matrices $A=\begin{pmatrix}
1 &4-a-a^2 \\
2 & -1
\end{pmatrix}$ and $B=\begin{pmatrix}
-a-1 &3 \\
3 & -5
\end{pmatrix}$ may represent the ...
0
votes
0answers
107 views
Quadratic Space Split over $\mathbb{R}$ implies Split over $\mathbb{Z}$?
Let $W \simeq \mathbb{Z}^{2n+1}$ be a free abelian group of rank $2n+1$ equipped with a symmetric nondegenerate bilinear form $\langle -, - \rangle \colon W \times W \to \mathbb{Z}$ of unit ...
1
vote
1answer
30 views
Vectorizing blocks of bilinear form
Math genius here. I have this problem of vectorizing the block matrix of bilinear entries as follows:
$$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{...
0
votes
1answer
39 views
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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0answers
21 views
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 \...
1
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1answer
50 views
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-...
0
votes
1answer
38 views
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 ...
0
votes
1answer
15 views
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 ...
1
vote
0answers
36 views
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?...
0
votes
1answer
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 ...
0
votes
0answers
32 views
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:
...
0
votes
1answer
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 ...
0
votes
0answers
38 views
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 ...
0
votes
0answers
34 views
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 ...
