# 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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### How to prove that $B$ is non-degenerate? does the question has typo?

Here is the question I want to prove: Let $V = M_2(\mathbb{R})$ be the space of all $2\times 2$ matrices over $\mathbb{R}.$ Show that $$B(X,Y) = det(X+Y) - det(X) - det(Y)$$ Where $X,Y \in V,$ is a ...
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### When will a complex representation of a semi-simple lie group, have an invariant hermitian form?

I know for compact groups, there is always one, and it can be made positive/negative definite. Does this remain true for non-compact groups, allowing for indefinite signature? I cannot think of an ...
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### Is there a name for this 'splitting' of an orthogonal structure into unitary and symplectic structures?

Suppose we have a (non-trivial) representation of some special orthogonal group $SO(p,q)$ over a real vector space $V$, I.e. the action of elements of $SO$ leave invariant a non-degenerate symmetric ...
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### The trace form is the unique non-deg. symmetric bilinear form such that ...

Let $L \vert K$ be a field extension. Recall the various definitions of the trace map $Tr_{L\vert K}:L\mapsto K$ (via matrices, minimal polynomials, field embeddings). With that, one gets the trace ...
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### Isomorphism between a vectorial space and its dual using bilinear forms.

If $g$ is a non degenerate symmetric bilinear form over a v.s. $V$ of finite dimension I want to find an isomorphism between $V$ and its dual in the more natural way possible. I thought about defining ...
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### Lattice embeddings in $\mathbb{Z}^n$

I have a certain class of lattices, and given $\Lambda$ in this class I would like to find obstructions to the existence of a finite-index lattice embedding $\Lambda \rightarrow \mathbb{Z}^n$ for some ...
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### Is this linear mapping positive definite, symmetric, bilinear?

$V=\mathbb{R}^2$ and let be$s:V \times V \rightarrow \mathbb{R^2}$ and $s(x,y)=2x_1y_2+2x_2y_2$ Well I found out that the matrix is: $$\begin{pmatrix} 0 & 2\\ 0 & 2 \end{pmatrix}$$ i) It's ...
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### Is nondegeneracy of a bilinear form equivalent for both slots?

According to Wikipedia, a bilinear form $B$ on a (possibly infinite-dimensional) vector space $V$ is defined as nondegenerate if the map $$B^\flat:V\ni x\mapsto B(x,\cdot)\in V^*$$ is an isomorphism ...
We want to find when a quadratic form depending on a parameter $\alpha$ on $\mathbb{R}^3$ is an inner product. It is quite obvious and quick to deal with the problem by using Sylvester, BUT we want to ...
Let $V$ be a vector space over a field $k$. The definition of quadratic form on nLab is a map $q:V\to k$ such that $q(tv)=t^2q(v)$ for all $t\in k,v\in V$, and such that the map \$(v,w)\mapsto q(v+w)-q(...