Tagged Questions

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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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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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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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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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 $\... 0answers 8 views There exists another bilinear symmetric map which is a multiple of$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 \}$.... 2answers 29 views $F$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 \} $. ... 0answers 56 views Is the converse to this theorem true? In the book that I'm reading there is this one theorem which states. Let V be a finite-dimensional vector space over a field F not of characteristic two. Then every symmetric bilinear form on V is ... 0answers 45 views Problem involving Bilinear Forms from Micheal Artin's book. The question is: Let T be a linear operator on$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$is an orthogonal ... 2answers 57 views signature of the topological manifold$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$is a compact 24-dimensional manifold ... 1answer 26 views questions about a proof of a theorem about symmetric nondegenerate bilinear forms Let$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^-...
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 ...