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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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 ...
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36 views

If $A \succeq B$ is it true that $B^{-1} \succeq A^{-1}$

If $A$ and $B$ are two positive definite matrices such that $A - B$ is nonnegative definite, is it true that $B^{-1} - A^{-1}$ is positive definite? The doubt came to me when working with confidence ...
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Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
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Linear Algebra, Quadric form

Question from an exercise Let $V$ be a vector space over a field $F$ with $charF\neq2$. If $\varphi,\psi\in V^{\vee}$ are linear functionals, we will define $\varphi\cdot\psi \colon V \rightarrow ...
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Trace of symmetric positive semidefinite matrix when diagonalized (as a bilinear form) in a non-orthogonal basis

Let $\mathbf{S}$ be symmetric positive semidefinite matrix (i.e. one with all eigenvalues real and non-negative). Then there is an orthogonal matrix $\mathbf{U}$ (with its columns forming an ...
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2answers
50 views

$\ker(A)=\text{Im}(A^*)^\perp$

How do I show that $\ker(A)=\text{Im}(A^*)^\perp$ for any square matrix $A$. I have done this problem before with the linear operator $T$ on a hermitian space but I can't seem to apply what I have ...
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Linear Algebra, Quadric Form, Bilinear Form

I have a question from an exercise. So $V$ is a vector space with quadric form $q:V\to \mathbb{R}$ . I have to prove that if the exists $u$, $v$ in $V$ such that $q(v)>0$ and $q(u)<0$ then ...
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54 views

If $x^T\!Ay=0$ for all $x,y $ in $ \mathbb{R}^n$ then $A=0$

Is it true that if $x^T\!Ay=0$ for all $x,y $ in $ \mathbb{R}^n$ then $A=0$. If so how do I justify this statement?
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Conceptual Question on different representations of Hyperplanes, Higher Standpoint, Coordinate-free

In a vector space $V$ over some field $F$ a hyperplane is the kernel of some linear transformation $T : V \to F$, i.e. the kernel of an element of the dual space (this could be taken as the definition ...
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33 views

Is there a closed form solution for slope lines of bilinear function?

Given a bilinear function $f(x,y) = a + bx + cy + dxy$, is there a closed form solution for a slope line passing through point $(x_0, y_0, f(x_0, y_0))$? It can exclude degenerate cases, e.g. $b = c = ...
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How to maximize this function

We are in an euclidian space, and we have to maximize the quadratic form : $x\in B\rightarrow (x|u) (x|v) $where $u$ and $v$ are two given vectors, and $B=\{x:||x||\leq1\}$ I don't find where i have ...
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27 views

Bilinear form over finite space

let $I$ be an ideal in local noetherian domain $(R,m)$ such that $I/I^2$ and $m/m^2$ are $R/m$ vector spaces , is there any nondegenerated symetric bilinear form on this spaces or the relation between ...
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Prove that a function $F$ is bilinear.

This time I brought a different problem. I'm starting the study of bilinear forms and I came across this question. This is probably simple, however still confuses me a bit. I must prove that ...
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23 views

How do i find a signature of a quadratic form? Also how do i represent a quadratic form as a sum/difference of squares?

For example given $(x,y,z,t) = xy+ y^2+ yz+z^2+zt$ How do i represent it as a sum and difference of squares (i.e. in the form $\sum a_iA_i^2$) and how do i find its trace? Or if i have a quadratic ...
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Non-central proper normal subgroups of unitary groups over fields

Short version: Can someone give an example of an anisotropic Hermitian form over a field such that its corresponding projective unitary group is not simple? Let $F$ be a (commutative, associative, ...
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If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that

If $f:\mathbb{R}^n \times \mathbb{R}^m \to \mathbb{R}^p$ is a bilinear function, then how to show that $$\lim\limits_{(h,k) \to (0,0)} \dfrac{|f(h,k)|}{|(h,k)|} = 0$$.
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Question about bilinear form [duplicate]

Prove that every bilinear form $f:\mathbb R^n \times \mathbb R^n\rightarrow \mathbb R$ has a basis $\{v_1,\ldots,v_n\} \subset \mathbb R^n$ such that $f(v_i,v_j)=-f(v_j,v_i)$ for every $i\neq j$. I ...
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Exam question: need to prove that for every bilinear form there's a basis that according to it, the bilinear form matrix is anti-symmetric [duplicate]

I need to prove that for every $$f:R^{n}\times R^{n} \to R $$ bi-linear form there is a basis $\{v_1,...v_n\}$ such that if $i$ is different from $j$ then $$ f(v_{i},v_{j}) = -f(v_{j},v{i}). $$ ...
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Prove that for every $f$, a bilinear form, there exists a basis ${v_1,…,v_n}$ so that $f(v_i,v_j) = -f(v_j,v_i)$

I didn't want to bloat the title, i'll also add that the basis is in ${\mathbb{R}^n}$. This question seems kind of easy, but its from a test so I assume there is a catch here somewhere. I tried to ...
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Generalized Poincaré Inequality

In my numeric script it states for a bounded domain $\Omega \subset\mathbb{R^n}$ and any map $0 < a_0 \leq a(x) \leq a_1 < \infty $ for $x\in \Omega$ it exists a $\gamma>0$ such that the ...
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Find the signature of $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$

In $\mathbb{R}^n$ let $Q(x_1,\ldots,x_n)= \sum_{i,j=1}^{n} a_ia_jx_ix_j$ quadratic form. $a:=(a_1,\ldots,a_n)\neq0$ $\in \mathbb{R}^n$ find the signature of $Q$
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Is Cea's lemma sharp?

Given a problem in weak formulation $$ \begin{align} \text{find $u\in V$ s.th. for all $v\in V$} \\ a(u,v) = f(v) \end{align} $$ with bilinear form $a:V\times V\rightarrow\mathbb{R}$, bounded with ...
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Homework: canonical form of quadratic form

X=(x,y,z) Q(X) = $x^2 + 4xy + 6xz + 3y^2 +8yz +5z^2 $ I got by using completing the square method: Q(X) = $(x+2y+3z)^2 - (y+2z)^2$ so as I learned now I do: $u = x+2y+3z$ $v = y+2z$ $w = 0 $ ...
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Find canonical form of bi-linear form on polynomials

$V$ is vector space of polynomials in degree less or equal than $2$, we define the bi-linear form: $f(p,q) = p'(-1)q(2$) where $p$ and q are polynomials from $V$. I need to find the canonical form ...
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Scalar / Dot product

I have a simple question about Scalar / Dot product. (http://en.wikipedia.org/wiki/Dot_product) Say f is a bilinear form. I have to tell if f defines a dot product. I didn't understand what I should ...
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Prove that $A^{t}A$ is positive definite

$A$ is an invertible matrix over $\mathbb{R}$ (nxn). Show that $A^{T}A$ is positive definite. I looked up for it and found this two relevent posts but still need help. positive definite and transpose ...
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Structure of $G$-invariant bilinear forms over (finite?) fields

I have a question about the structure of $G$-invariant bilinear forms. Let $G$ be an arbitrary finite group and $\mathbb{F}_q$ a finite field such that $2|G|$ is not divisible by the characteristic of ...
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4answers
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Tensor product in multilinear algebra

In the book by Halmos ($FDVS$) the tensor product of two vector spaces U and V is defined as the dual of the vector space of all the bilinear forms on the direct sum of U and V. Is there a generalised ...
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Find an orthonormal basis of a particular bilinear form

Let $V=\mathbb{R^3}$. Find an orthonormal basis in which the bilinear form with matrix $A$: $\begin{pmatrix} 2 & -2 & 0 \\ -2 & 1 & -2 \\ 0 & -2 & 0\end{pmatrix}$ has a ...
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1answer
25 views

Condition for continuity of bilinear form

In my numeric script there is a unproved theorem, saying that a bilinear form $a \colon V\times V \to \mathbb{R}$ on a normed vector space $V$ is continuous if and only if $$|a(v,w)| \leq c \, \|v\| ...
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Centralizing a maximal flag in a symplectic group

Short version: I'm confused about maximal totally-isotropic flags versus maximal flags: do they have the same centralizer in the classical group? Let $F$ be a field, $V$ be a finite dimensional ...
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Bilinear transformation and eigenvalues

I have a proof to do and I am stuck on proving that if there exist a matrix $A$ with eigenvalue $\lambda_i$ and $B$ with eigenvalues $\mu$ such that $A = (B+I)(B-I)^{-1}$ then we have ...
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Find basis for $P_0^\perp \subset P_4$ and $\ker (f \mapsto f(0))$ in $P_4$

Let $P_n \subset \textrm{Map}(\mathbb{C},\mathbb{C})$ the space of polynomial maps $\mathbb{C} \to \mathbb{C}$ with degree $\le n$. We define $\langle f,g\rangle := \int_{-1}^1 f(t)\overline{g(t)}dt$. ...
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Diagonalizng a bilinear form

Question a. we mark $\mathbb{R}_2[x]$ as the polynomial space of degree $ \le 2$ over the real field $\mathbb{R}$. $\xi :\mathbb{R}_2[x] \times \mathbb{R}_2[x] \to \mathbb{R}$ $$\xi(q,p) = ...
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1answer
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Matrices in Linear Algebra

Let: $ u: R^2 --> R^3$ be defined by: $$ u(x,y)=(x+2y, 2x-y, 2x+ 3y)$$ Give the matrix $M[u]$ in the canonical base of its definition space. This question might seem sort of stupid, but it was ...
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45 views

Show that $B$ represents an inner product.

Let $V$ be a vector space over $\mathbb{R}$ with $\dim V = n$ Let $B = (b_{ij})$ be an $n \times n$ diagonal matrix which represents a bilinear form on $V$ with respect to a basis of $V$ , ...
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Find basis such that bilinear forms attain normal form

Let $g :=$ symmetric bilinear form : $g(v,w) := \omega(v,Jw)$ with $J \in \textrm{O}(V,g)$ and $\omega := $ skew-symmetric bilinear form : $\omega(v,w) := g(v,Jw)$ with $J \in \textrm{Sp}(v,\omega)$. ...
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Norm of an operator defined on sequence spaces

Consider the sequence space $\ell_r$ defined by $$\ell_r=\left\{x=(x_n)_{n=1}^{\infty}:x_n\in\mathbb{R}\text{ and }\sum_{n=1}^{\infty}|x_n|^r<\infty\right\}.$$ Let $2\leq p,q<\infty$ such that ...
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Calculating the signature of a matrix

The task is the following: Consider $\mathbb{R}^2$ equipped with the canonical dot-product $\langle \cdot , \cdot \rangle$, and also the symmetrical bilinear form $$\beta(u,v) := \left\langle u,\ ...
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Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...
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Show isomorphism $W_1 \hookrightarrow V \twoheadrightarrow W_2$

Let $\langle , \rangle$ be a non-degenerate bilinear form with the signature $(p,q)$ on a real vectorspace $V$ and $W_1, W_2$ subspaces, such that the restriction $\langle , \rangle |_{W_i}$ is ...
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Show skew-symmetric, non-degenerate bilinear form $((a, \varphi),(b, \psi)) \mapsto \langle(a, \varphi),(b, \psi) \rangle := \varphi(b)-\psi(a)$

Let $W$ be a finite dimensional $K$ vector space and $W^*$ its dual space. For $V := W \oplus W^*$ the mapping $$ V \times V \to K,((a, \varphi),(b, \psi)) \mapsto \langle(a, \varphi),(b, \psi) ...
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Show $\langle , \rangle |_W$ non degenerate $\implies$ $\langle , \rangle |_{W^\perp}$ non degenerate

Let $W \subset V$ be a subspace and $\dim V < \infty$. If $\langle , \rangle$ and the restriction $\langle , \rangle |_W$ are non degenerate, then $\langle , \rangle |_{W^\perp}$ is non degenerate ...
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Sesquilinear forms seen as bilinear maps

Let $V$ be a complex vector space. A sesquilinear map (or conjugate-linear in the first variable and linear in the second) on a complex vector space $V$ is a map $f: V \times V \rightarrow \mathbb{C}$ ...
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Symmetric matrix congruency

There is a sentense that says that every symmetric matrix is congruent to a diagonal matrix. I've been trying to find the congruent matrix and the transition matrix for the following: $$ ...
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1answer
11 views

If a symmetric bi-liear form is positive, then $a_{11}\cdot a_{nn}>a_{n1}\cdot a_{1n}$

I need to prove that if a bi-linear form is symmetric and positive, then it's representative matrix $A=(a_{ij})_{i,j=1}^n$ satisfies: $$a_{11}\cdot a_{nn} > a_{n1}\cdot a_{1n}$$ I've tried for ...
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38 views

Generalisation of Gramian determinant

i'm wondering about those facts of basic linear algebra: if you have $n$ vectors $x_1,...,x_n \in \mathbb{R}^n$, you can easily test their linear dependance by computing their Gramian Matrix $M$ whose ...
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40 views

Show that a map is a continuous bilinear form on $H^1(0,1)$ space

Let $u,v \in H^1(0,1) = \{f : (0,1) \longrightarrow \mathbb{R}, f,f' \in L^2(0,1) \}$, show that $$a(u,v) = \int_0^1 (u'v' + uv)\; dx$$ is a continuous bilinear form.
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Linear systems and bilinear forms

Given a general linear system $Ax = y$ and the bilinear form $z(x,y) = y^T Ax$, what are the links between these two mathematical objects? Thanks. EDIT: Original question is too general and ...