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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The map $f$ is degenerate or non-degenerate?

Let denote by $M_{3,2}(\mathbb C) $ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
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Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
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Geometric meaning of Equation

As a part of my linear-algebra exam preparation, I am going through the surface equation and quadratic-bilinear form usage in my book which is a part we haven't really went through and left to explore ...
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Matrices A and B have same number of positive and negative Eigenvalues $\Leftrightarrow$ $\exists \: C $ so that $C^t AC=B$?

Let $A,B \in M(n,n;\Bbb R)$ symmetric. Look at the two statements: $A$ and $B$ have the same number of positive and negative Eigenvalues $\exists \: C \in GL_n (\Bbb R )$ with $C^tAC=B$ I have to ...
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Using an arbitrary Hermitian matrix to define a Hermitian form, then proving a related subspace property

Let $H$ be an invertible $n$ by $n$ Hermitian matrix. Use $H$ to define a Hermitian form, $[ \cdot , \cdot ]$ which takes two column vectors $x,y \in \mathbb{C^n}$ and has the definition $[x,y] = {\...
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Describe an equation geometrically

Finishing the last few stuff left for my end-term semester exams on Linear Algebra II, I bumped across a collection of identical exercises, posting one below : Describe geometrically, giving as much ...
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inner product and bilinear mappings

I understand that the inner product of two vectors and its properties. However I do no quite understand bilinear mappings. What is the relationship between inner products and bilinear mapping? and ...
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Show that the bilinear form is well-defined

Let $V$ be a vector space over $K$, $\langle-.-\rangle$ a symmetric bilinear form on $V$ and $T \subseteq V$ with $T :=$ {$v \in V$ | $\langle v, u\rangle = 0$ $\forall v \in V $}. I have to show ...
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Is bilinear form V-elliptical?

I have a following bilinear form: $$ \int^1_0 u'v' dx + u(1)v(1) $$ I'd like to prove, that it's V-elliptical, so that: $$ a(v,v) \geq m \cdot ||v||_{2,1}^2, \forall v \in H^1_V(\Omega) $$ My ...
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Prove bilinear form is nondegenerate

This is from Linear Algebra, an Introductory Approach - Charles Curtis Let $V$ be a finitely generated vector space over $F$ with basis $\{v_1, ..., v_n\}$. Let $A = (\alpha_{ij})$ be a fixed $n$ by $...
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Prove or disprove: quadratic form $Q(v):=\beta(v,v) $ (biliniarform) is nondegenerate if $\beta$ is.

Let char$K \not = 2$ and $\beta$ be a bilinearform (not necessarily symmetric) on a $K$-vectorspace $V$. Let $Q$ be defined by ($v, w \in V$ arbitrary)$$Q(v) = \beta (v,v)$$ I've shown that $Q$ is a ...
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Proof the existence of a certain bilinear form on the vector space V/T

Let $\mathbf V$ be a vector space (over a field $\mathbf K$) together with a symmetric bilinear form <-,->, and let $\mathbf T $ $\subseteq$ $\mathbf V$ be the orthogonal complement (since I'm not ...
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The matrix of a bilinear form

Let the dimension of the space $V$ over the field $\mathbb{R}$ is odd, and $B$ is the matrix of some non-degenerate bilinear form $\beta$ on $V$. Can the matrix $B$ be the matrix of a bilinear form $-\...
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A bilinear diophantine problem

Suppose we know $a,b,c,d,e,f,m\in\Bbb Z$ in $$(a^2c+b^2d)y+ab(vy)+(a^2e+b^2f)v=m$$ how do we find $v,y\in\Bbb Z$?
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Did I make mistakes? Bilinear form, generator, strange relation

I have a question about functional analysis and operator theory. Definition Let $(H,(\cdot,\cdot)_{H})$ be a real Hilbert space and $D$ be a dense subspace of $H$. Let $(\mathcal{E},D)$ be ...
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Proof of Positive definiteness of a scalar/dot product

$ n, d \in \mathbb R , V = \{f ∈ R[X] \mid \deg(f) ≤ n\}$ and $x_0,\ldots,x_d \in \mathbb R$ are distinct. Prove $ \langle f,g\rangle := \sum_{i=0}^d f(x_i)g(x_i)$ positive definite is, when $d ≥ n$. ...
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How could I find an orthogonal basis of this bilinear form f?

Where $f : \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}$ corresponding to the quadratic form $q : \mathbb{R}^3 \rightarrow \mathbb{R}$, $q(x,y,z) = x^2 + 2xy + y^2 + 2yz + z^2$ I found ...
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Show $\{\beta (v,w) =0 \Leftrightarrow \beta(w,v)= 0\} \:\:\:\: \Rightarrow \:\:\:\: \{\beta(w,u)\beta(u,v) = \beta (v,u) \beta (u,w) \}$

$\beta$ is a bilinear form on a $K$-vectorspace $V$. Now i have to show the following: if $\forall$ $v,w \in V$ $$\beta (v,w) =0 \Leftrightarrow \beta(w,v)= 0 $$ then $\forall u,v,w \in V$ $$\beta(...
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Find an orthogonal base of a bilinear form on a field of characteristic 2

Let $K$ be a Field of characteristic $2$. On $V=K^2$ the symmetric bilinearform $\beta (x,y) = x_1y_2+x_2y_1 $ is defined. Now i have to either find an orthogonal base of $V$ or show that such a ...
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Sylvester's argument for bilinear functions

Let $V$ be a vector space of dimension $n$ and let $b:\colon V \times V\to \mathbb{R}$ be a symmetric bilinear function. Sylvester's theorem says that there exists a basis of $V$ with respect to ...
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Proving that for every $2$-form $\omega$, a basis of $V^*$ exists, so that $\omega$ can be written in a certain way

Let $\omega$ be an alternating $2$-form on an $n$-dimensional $\mathbb{R}$ vector space $V$, where $\omega$ isn't $0$ everywhere. I want to prove: There exists a basis $(\alpha_i)_{1 ≤ i ≤ n}$ of $V^*...
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How Coulomb Gauge guarantees uniqueness in regard to Lax-Milgram Lemma, curl-curl problem

The Lax-Milgram lemma gives insight on existence and uniqueness of a PDE of the type $$ a(u,v)=f(v) $$ Positive definiteness and coercivity are required for the bilinear form $a(u,v)$. In the curl-...
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Upper bound of a bilinear form

Suppose I have a form $|X^TBY|$ where $X \in R^n, Y \in R^m$ and $B \in R^{n \times m}$ is a matrix whose elements are bounded. Is there an upper bound for the whole expression of the form $|X^TBY|\le ...
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How does one prove that a bilinear form is positive definite?

$\mathrm{V}$ is a $\mathbb{Q}$ - vector space with base $\mathrm{B} = (v_1,....v_n)$. A symmetric bilinear form is given with: $$\mathrm{F}(v_i,v_j) = \{ \text{1 for i = j}, \frac{1}{2} \text{ for i =...
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Proof of $\text{rank}(A) = \text{dim}(V) - \text{dim}(\text{radical}(V))$

Let $V$ be an $n$ dimensional vector space with a symmetric bilinear form induced by the matrix $A$, that is, $a_{ij} = (v_i, v_j)$ where $v_i$ are a basis of $V$. I am stuck trying to show that $\...
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Symmetric, Antisymmetric, and Alternating Bilinearforms form a vector subspace

I have to show that the space of symmetric, the antisymmetric and the alternating bilinear forms each form a vector subspace of the space of all bilinear forms $\operatorname{Bil}(V,K)$ with $V$ being ...
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Symplectic form on $\mathbb R ^{2n}$

What are all symplectic form $\omega$ on $\mathbb R^{2n}$. Where, a ''symplectic bilinear form'' on $\mathbb R^{2n}$ is . a bilinear form: a map $\omega: \mathbb R^{2n}\times \mathbb R^{2n}\to \...
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signature of a bilinear form

This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8. Lemma: let $E$ be an n-dimensional real vector space endowed with a ...
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bilinear form and positive definiteness

Let $B$ a symmetric bilinear form on an $n$ dimensional vector space $E$ with signature $(n-1,1)$. Then there exists a hyperplane $H$ in $E$ in which $B$ is positive definite. How to prove this? Is ...
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Positive definiteness of a bilinear form implies symmetry?

In the Wikipeda article about positive definite bilinear forms, there is the line It turns out that the matrix $M$ is positive definite if and only if it is symmetric and its quadratic form is a ...
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Sesquilinear Forms

I was trying to solve some exercises related to sesquilinear forms: Let V be a C-vector space (C - complex numbers) Prove that the set $\mathcal{S}(V)$ of sesquilinear forms on V is a vector ...
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Weak formulation of non-local Neumann problem

Consider the following probleblem: $$ -\Delta u +a(x)\int_{\Omega}b(z)u(z)dz = f \qquad \text{in $\Omega$} $$ $$ \partial_{\nu}u=0 \qquad \text{in $\partial\Omega $} $$ where $$\Omega\quad \text{...
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If $W$ is proper subspace of a real finite dimensional $V$, $\exists v\in V $ s.t. $H(w,v)=0 \forall w \in W$

$H(x,y)$ is a bilinear form. I have tried to do something similar to standard inner product in $\Bbb{R}$ such as Gram-Schmidt process, orthogonal complement...But they don't work since we don't know ...
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Orthogonal projection of skew-symmetric form

It is a question from the book Algebra by Michael Artin: 8.8.2 Let W be a subspace on which a real skew-symmetric form is nondegenerate. Find a formula for the orthogonal projection $\pi:V\to W$
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Prove the positive definiteness of Hilbert matrix

This is so called Hilbert matrix which is known as a poorly conditioned matrix. $$ A = \left(\begin{matrix} 1 & \frac{1}{2} & \frac{1}{3} & ... & \frac{1}{n} \\ ...
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There is no positive definite bilinear form on $V$

Let $V$ be a $\mathbb C$-vector space, $\dim V>1$. Then for any bilinear form $\phi$ there is $v\in V$ s.t. $\phi(v,v)=0$ (so there is no positiv definite bilinear form on $V$) If one takes the ...
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Perfect Pairing, non-degeneracy and dimension.

On this wikipedia entry https://en.wikipedia.org/wiki/Bilinear_form#Different_spaces it tells us that if $B: V \times W \to K$ is a bilinear map, then In finite dimensions, [a perfect pairing] is ...
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Orthogonal complement of orthogonal complement of U equals U

Let $V$ be of finite dimension over some field $F$. Let $ξ\in T_2(v)$ symetric bilinear form. Let $U\subset V$. suppose $ξ|_U$ is not degenerate, is it necessarily $(U^⊥)^⊥=U$ ? my approach: I ...
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Weierstrass theory of elementary divisors

I'm taking a history of algebra course and I can't seem to figure out Karl Weierstrass' Theory of Elementary Divisors. It says he began by considering two quadratic forms $\phi = \sum_{i,j=1}^n a_{ij}...
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Find the orthogonal complement with respect to $\phi$

Given is a symmetric bilinear form $\phi:\mathbb R^4\times\mathbb R^4\to \mathbb R$ with $\phi(v,w)=v^{T}Aw$ and $A=\begin{bmatrix}1&1&1&1\\1&2&2&2\\1&2&-2&0\\1&...
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Hermitian Pairings from Positive Functionals

Let $A$ be $*$-algebra and $\phi:A \to {\mathbb C}$ a positive linear functional, that is, one for which $\phi(aa^*) \geq 0$, for all $a \in A$. When does it hold that a symmetric sesquilinear form, i....
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Root systems and the possible angles between roots.

On page 4 of these notes by John Dusel (http://math.ucr.edu/~jmd/Root_Systems.pdf) it reminds us that if we have a symmetric positive bilinear form (pageg 2) we can define an angle between vectors $\...
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There exists another bilinear symmetric map which is a multiple of $F$

This question is a continuation of the following question. So we have $F: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ a bilinear symmetric map, and $K = \{ v \in \mathbb{R}^n \mid F(v,v) = 0 \}$....
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$F$ is indefinite if and only if $K$ is not a subspace

I get the following linear algebra problem in my class. Let $F: \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ be bilinear and symmetric, and let $K = \{ v \in \mathbb{R}^N \mid F(v,v) = 0 \} $. ...
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Is the converse to this theorem true?

In the book that I'm reading there is this one theorem which states. Let V be a finite-dimensional vector space over a field F not of characteristic two. Then every symmetric bilinear form on V is ...
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Problem involving Bilinear Forms from Micheal Artin's book.

The question is: Let T be a linear operator on $V=R^n$ whose matrix $A$ is a real symmetricmatrix. a) Prove that $V$ is the orthogonal sum $V=(ker T)\oplus(im T)$ b) Prove that $T$ is an orthogonal ...
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signature of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ (using homology, cohomology)

I want to prove that the signature $\operatorname{sig}(M)$ of the topological manifold $M= \mathbb{C}P^6\times \mathbb{C}P^6$ is nonzero. First of all, $M$ is a compact 24-dimensional manifold ...
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questions about a proof of a theorem about symmetric nondegenerate bilinear forms

Let $V$ be a finite dimensional $\mathbb{R}$-vector space ($\dim_\mathbb{R} V=n$) and $\varphi:V\times V\to\mathbb{R}$ a symmetric nondegenerate bilinear form. Then there exist subspaces $V^+$ and $V^-...
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Bilinear (non-convex) objective and linearized (big-M) constraints

My original problem is a MIQCQP. The bilinear terms in the constraints are products of binary and continuous variables and can be linearized using big-M. The bilinear terms in the objective function ...
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Simultaneous Diagonalization of two bilinear forms

I need to diagonalize this two bilinear forms in the same basis (such that $f=I$ and $g$=diagonal matrix): $f(x,y,z)=x^2+y^2+z^2+xy-yz $ $g(x,y,z)=y^2-4xy+8xz+4yz$ I know that it is possible ...