[Community Question] Linear-algebra: Unitary Central Character by Schur's Lemma

One of our user asked:

Consider an irreducible smooth representation $\pi$ of the group $G=GL_n(\mathbb{Q}_p)$ with center $Z$. Does there exist a unitary central character for $\pi$?

More precisely, is there a (quasi-)character $\omega: G \to \mathbb{C}^{\times}$ such that $\pi \otimes \omega$ when restricted to the center $Z$ is a unitary character for $Z$? I find this result casually stated in many references, where they say it follows from Schur's lemma. But I am unable to see it directly from Schur's lemma.

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[Community Question] Linear-algebra: non-negative matrix satisfying two conditions

One of our user asked: A real matrix $B$ is called non-negative if every entry is non-negative. We will denote this by $B\ge 0$ . I want to find a non-negative matrix $B$ satisfying the following two conditions: (1) $(I-B)^{-1}$ exists but not non-negative. Here $I$ is the identity matrix. (2) There is a non-zero and non-negative vector $\vec{d}$ such that $(I-B)^{-1}\vec{d}\ge 0$ . I tried all the $2\times 2$ matrices, but it did not work. I conjecture that such a $B$ does not exist, but don't know how to prove it.

Calculus: Intersection of hyperplanes in $\mathbb{R}^5$

Consider the $4$ hyperplanes in $\mathbb{R}^5$ given by the equations $$x_1-x_2+x_3+x_4+2x_5=0$$ $$2x_1-x_2+6x_3+2x_4+6x_5=0$$ $$3x_1-3x_2+3x_3+4x_4+7x_5=0$$ $$x_1+x_2+9x_3+x_4+6x_5=0$$ Let $V \leq \mathbb{R}^5$ be the intersection of these hyperplanes. Find a basis for $V$ .

[Community Question] Geometry: The limit about the line connecting the intersection of a circle and the $y$-axis and the intersection of the shrinking circle and a fixed circle

One of our user asked: There is a fixed circle $C_1$ with equation $(x - 1)^2 + y^2 = 1$ and a shrinking circle $C_2$ with radius $r$ and center the origin. $P$ is the point $(0, r)$ , $Q$ is the upper point of intersection of the two circles, and $R$ is the point of intersection of the line $PQ$ and the $x$ -axis. What happens to $R$ as $C_2$ shrinks, that is, as $r \to 0^+$ ? (The figure is made with GeoGebra ) In order to solve this problem, I made a script using GeoGebra in which the circle $C_2$ is a dynamic one whose radius $r$ can be adjusted with a slider. As I set $r \to 0^+$ , the figure seems to suggest that $R \to (4,0)$ . In particular, this is the state with $r = 0.001$ , in which $R$ is reported to be $(3.9999997523053,0)$ : However, I would like to find out a way to prove (or disprove, though unlikely) my guess that $$\lim_{r \to 0} R = (4,0).$$ But I have little idea. Any help would be appreciated.

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