While attempting to answer a question here (namely, the finite dimensional case of the title question: Prove that if $\lambda$ is an eigenvalue of $T$, a linear transformation whose matrix representation has all real entries, then $\overline{\lambda}$ is an eigenvalue of $T$), I noticed the asker did not specify a finite dimensional vector space. Though […]

Remark: I give much background because it might significantly help to find an idea how to generate a solution I’m analyzing properties of a certain infinite matrix $U$, for whose columns we have the generating functions $$f_c(x) = (\exp(x)-1)^c \qquad \qquad c=-\infty…+\infty $$, assuming $c$ from $-\infty$ to $+\infty$ (This is the type of Carleman-matrices […]

Let $x \in l_2$ and $J(x) = \sum_{k = 1}^{+\infty} x_k x_{k + 1}$. Find $DJ(u)$ and $D(DJ(u))$. Attempted solution Since $x \in l_2$, then $\sum_{k = 1}^{+\infty}x_k < \infty$. Another fact: $\|x\|_{l_2} = (\sum_{k=1}^{+\infty}x_k^{2})^{\frac{1}{2}}$. By the definition of Frechet derivative: $$ \lim_{h\to0} \frac{|J(u + h) – J(u) – (DJ(u))(h)|}{\|h\|_{l_2}} = \lim_{h\to0} \frac{|\sum_{k = 1}^{+\infty}(u_k […]

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: $\qquad \qquad P_{n=5}=\small \begin{bmatrix} 1 & 1 & 1 & 1 & 1 \\ . & 1 & 2 & 3 & […]

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