Articles of bilinear form

Lagrange diagonalization theorem – what if we omit assumption about the form being symmetric

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 […]

possible signatures of bilinear form on subspaces

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 (+,+,+), […]

Give an example of a discontinuous bilinear form.

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.

Finding the symplectic matrix in Williamson's theorem

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?

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..

Show that a Bilinear form is Coercive

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 […]

Rank $1$ bilinear form is a product of two linear functionals on a finite dimensional vector space.

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 […]

Trace of a bilinear form?

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}( […]

Bilinear form with symmetric “perpendicular” relation is either symmetric or skew-symmetric

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 […]

Simultaneous diagonlisation of two quadratic forms, one of which is positive definite

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 […]