I know that for every symmetric form $f: U \times U \rightarrow \mathbb{K}$, char$\mathbb{K} \neq 2$ there exists a basis for which $f$’s matrix is diagonal. Could you tell me what happens if we omit assumption about $f$ being symmetric? Could you give me an example of non symmetric bilinear form $f$ which cannot be […]

Suppose to have been given the signature of a symmetric bilinear form on a finite dimensional vector space. Is there a general rule to get all the possible signatures of the restriction to subspaces of codimension 1? For instance, I know that if the signature is (-,+,+,+) all subspaces of codimension one have signatures (+,+,+), […]

I’m having a hard time with this, I asked a tutor and he couldn’t think of an example either. The bilinear form on a vector space $V$ is a map $f: V \times V \rightarrow F$ that satisfies linearity in both variables.

tl;dr: How do I construct the symplectic matrix in Williamson’s theorem? I am interested in a constructive proof/version of Williamson’s theorem in symplectic linear algebra. Maybe I’m just missing a simple step, so here is what I know: Let us fix the symplectic form $J=\begin{pmatrix} 0_n & 1_n \\ -1_n & 0_n\end{pmatrix}$. Recall: Theorem: Let […]

Is every symmetric bilinear form on a Hilbert space a weighted inner product? i.e. can I write that $b(u,v) = (wu,v)_H$ for all $u, v \in H$? I am not sure about this. Maybe something to do with Riesz theorem..

I’m reading through Brezis’ book on functional analysis, Sobolev spaces and PDE, and I’m having trouble showing that the Bilinear form: $a(u,v) = \int_{0}^{2}u’v’dx+\left(\int_{0}^{1}u dx\right) \left(\int_{0}^{1}v dx\right)$ is coercive.(Problem 8.25 #2). The book offers the following hint: (Argue by contradiction and assume that there is a sequence $(u_n)$ in $H^1$ such that $a(u_n,u_n)\rightarrow 0$ and […]

Let $\mathbb{f}$ be a non-zero bilinear form on a finite dimensional vector sppace $V.$ Then have to show that $\mathbb{f}$ can be expressed as a product of two linear functionals i.e., $\mathbb{f}(\alpha, \beta)=L_1(\alpha)L_2(\beta)$ for $L_i \in V^*$ iff $\mathbb{f}$ has rank $1.$ I proved that if $\mathbb{f}$ is product of two linear functional then its […]

I’m just a beginner of differential geometry, so please forgive me if this is nothing but a silly question or I’m making a critical conceptual mistake. Let $\mathrm{I\!I}(X, Y)$ be the second fundamental form on an embedded $n$-manifold in the $(n+1)$-dimensional Euclidean space. Then the mean curvature $H$ is defined as the trace of $\mathrm{I\!I}( […]

Let $b$ be a bilinear form on a finite-dimension vector space $V$ (over a field with char $\neq$ 2) such that for each $x,y\in V$ one has $b(x,y)=0\Leftrightarrow b(y,x)=0$. Prove that $b$ is symmetric or skew-symmetric. The condition is equal to this: for every vector $x$ co-vectors $b(\cdot,x)$ and $b(x,\cdot)$ have the same kernels, so […]

Let $\varphi, \phi$ be quadratic forms on $V$ and suppose $\varphi$ is positive definite. I want to find a basis for V such that $\varphi$ and $\phi$ are both represented by diagonal matrices. My idea is to define an inner product $<,>: V\times V \rightarrow F$ where $<v,w> = \varphi(v,w)$. I know that if I […]

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