Do commutative matrices share the same eigenvectors?

Let two square matrices A and B represent linear operators on a vector space V over C. Suppose they are commutative. Then ABx=BAx,∀x∈V Then let ˜x be an eigenvector of B. Setting, x=˜x, we see that AB˜x=BA˜xAλ˜x=BA˜x,∃λ∈Cλ(A˜x)=B(A˜x)⟹A˜x=α˜x,∃α∈C So every eigenvector of B is also an eigenvector of A. ◼ MY QUESTIONS: Is this valid? If … Read more

${\rm sup}\ A\cap B = {\rm min}\ \{ {\rm sup} (A), {\rm sup}(B) \} $

Let $A,B\subseteq \mathbb{R}$ be a non-empty intervals and bounded from above. If $A\cap B\neq \emptyset $ prove that it is bounded from above and that $Sup(A\cap B)=min\{sup(A),Sup(B)\}$ $A,B$ are bounded from above therefore there are $M_1,M_2$ such that $a\leq M_1$ and $b\leq M_2$. for $x \in A\cap B$ $(x\in A \wedge x\in B)$, so $x\leq … Read more

Verification for this proof

Sorry guys about the verification questions but it’s near the end of the semester and I am very sheepish about making mistakes especially because real analysis is a very important course it’s only the underlying theory of calculus. This problem may seem rudimentary but we were given a sample test by our professor and it … Read more

Is ∑∞n=1ansin(nx)\sum_{n=1}^\infty a_n\sin(nx) converges on [ε,2π−ε][\varepsilon, 2\pi-\varepsilon]?

Let an, a sequence monotonically decreasing to 0. Consider ∞∑n=1ansin(nx) Is the series converges uniformly on [ε,2π−ε]? (ε>0) Basically we could use Dirichlet’s test. We want to show that ∑∞n=1sin(nx) is bounded. Indeed: ∞∑n=1sin(nx)=i2(∞∑n=1(eix)n+∞∑n=1(e−ix)n)≤i2(11−eix+11−e−ix)≤11−ei(2π−ε)<∞ BUT, clearly, g(π2)=∞∑n=1sinnπ2=∞ Where is the mistake? Answer We show that if an is monotonically decreasing, then the series ∞∑n=1ansin(nx) is … Read more

Proving 10\cdot n=010\cdot n=0 for all n\in\mathbb{Z}n\in\mathbb{Z} with n\geq 0n\geq 0 using strong induction

The question says 10\cdot n=0 for all n\in\mathbb{Z} with n\geq 0. Here is my proof by strong induction: Base case: 10\cdot0=0. Let k\geq 0, and suppose that for any m\leq k we have that 10\cdot m=0. Consider 10\cdot(k+1). The number k+1 can be written as m+l for some numbers 0\leq m,l\leq k. By the induction … Read more

rk(A2)=rk(B2)⟹rk(A)=rk(B)rk(A^2)=rk(B^2) \implies rk(A)=rk(B) is it true?

The original statement is this: given A,B matrices n×n, if A2 is “Left-Right equivalent” to B2 then A is LR equivalent to B (is it true or false?) I know that A is LR equivalent to B iff rk(A)=rk(B) so I decide to work with ranks. I think it’s true so I tried a proof … Read more

An algebra $A$ is a $\sigma$-algebra iff it’s closed under countable unions

An algebra $A$ is a $\sigma$-algebra if and only if $A$ is closed under countable increasing unions. Proof: Suppose $\{E_j\}_{1}^{\infty}\subset A$ and $E_1\subset E_2\subset \ldots$ Set $$F_k = E_k \setminus \big[\bigcup_{1}^{k-1}E_j\big] = E_k \cap \big[\bigcup_{1}^{k-1}E_j\big]^{c}$$ Then the $F_k$\s belong to $A$ and are disjoint, and $$\bigcup_{1}^{\infty}E_j = \bigcup_{1}^{\infty}F_k$$ Therefore, $\bigcup_{1}^{\infty}E_j\in A$ I am not sure … Read more

There is a best performer in a round robin tournament

At a social bridge party every couple plays every other couple exactly once. Assume there are no ties. If n couples participate, prove that there’s best couple in the following sense: A couple u is best if for every couple v, u beats v or u beats a couple that beats v. What I tried: … Read more

Let (x,y)(x,y) be the smallest solution ∈N+\in \mathbb N^+ of x2+xy−y2=0x^2+xy-y^2=0, then y−x

The book shows why x2+xy−y2=0 doesn’t have any solutions in N+: Let (x,y) be the solution with smallest x∈N+ of x2+xy−y2=0 (where y must be >x). Then (y−x,x) is also a solution, but with smaller first coefficient. Contradiction. How can I deduce that y−x is smaller than x? I understand that this is true for … Read more