There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is that, I have $2$ layers of atoms where $A$ is connectivity within the layer $1$ itself and $B$ is connectivity between layer […]

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ H^k(\mathbb{T}^2) \simeq H^k_{per}([-\pi,\pi)).$$ Here, $H^k$ denotes the Sobolev space $W^{k,2}$ equipped with the norm of your choice. I have two questions: 1) Is there an orthonormal basis $\lbrace v_n \rbrace_{n \in […]

Given a sequence of functions orthogonal over some interval, which satisfy Dirichlet boundary conditions at that Interval, can we construct a Sturm Liouville problem that gives these as its eigenfunctions? For example, if we take the sequence of scaled Bessel’s functions $J_n (\zeta_i x)$ for all positive integral values of $i$ where $\zeta_i$s are roots […]

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$. If $p \in V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. What are the eigenfunctions belonging to this eigenvalue? It is obvious that constant polynomials are eigenfunctions with eigenvalues […]

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