Most Popular
1500 questions
69
votes
9 answers
Do all square matrices have eigenvectors?
I came across a video lecture in which the professor stated that there may or may not be any eigenvectors for a given linear transformation. But I had previously thought every square matrix has eigenvectors.
sash
- 843
69
votes
7 answers
$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$ approximation
Is there any trick to evaluate this or this is an approximation, I mean I am not allowed to use calculator. $$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots}}}}}$$
user2378
- 1,081
69
votes
3 answers
The $\sigma$-algebra of subsets of $X$ generated by a set $\mathcal{A}$ is the smallest sigma algebra including $\mathcal{A}$
I am struggling to understand why it should be that the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$ should be the smallest $\sigma$-algebra of subsets of $X$ including $\mathcal{A}$. Let me try to elucidate my understanding of the…
Harry Williams
- 1,839
69
votes
3 answers
What is Haar Measure?
Is there any simple explanation for Haar Measure and its geometry? How do we understand the analogy between Lebesgue measure and Haar Measure? How to show integration with respect to Haar Measure? What do we mean by integrating with respect to…
Milan Amrut Joshi
- 1,215
69
votes
1 answer
How to prove this recurrence relation for generalized "rounding up to $\pi$"?
The webpage Rounding Up To $\pi$ defines a certain "rounding up" function by an extremely simple procedure: Beginning with any positive integer $n$ , round up to the nearest multiple of $n-1$ , then up to the nearest multiple of $n-2$ , and so on,…
r.e.s.
- 15,627
69
votes
4 answers
why geometric multiplicity is bounded by algebraic multiplicity?
Define The algebraic multiplicity of $\lambda_{i}$ to be the degree of the root $\lambda_i$ in the polynomial $\det(A-\lambda I)$ . The geometric multiplicity the be the dimension of the eigenspace associated with the eigenvalue $\lambda_i$ . For…
Jack2019
- 1,615
69
votes
7 answers
Geometric understanding of differential forms.
I would like to understand differential forms more intuitively. I have yet to find a book which explains how the use of the exterior product in differential forms ties into the geometrical significance of it. Most books briefly introduce the…
Markus
- 693
69
votes
2 answers
What does "communicated by" mean in math papers?
[This question involves mostly math papers, and may be relevant to graduate students learning to write and cite papers, although this is my only justification for this being a math question.] Usually papers start out with the title and then the…
user2055
69
votes
15 answers
Prove if $n^2$ is even, then $n$ is even.
I am just learning maths, and would like someone to verify my proof. Suppose $n$ is an integer, and that $n^2$ is even. If we add $n$ to $n^2$, we have $n^2 + n = n(n+1)$, and it follows that $n(n+1)$ is even. Since $n^2$ is even, $n$ is even. Is…
user79612
- 575
69
votes
4 answers
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on itself. I know that a categorical approach is…
Rachmaninoff
- 2,680
69
votes
8 answers
Good books on Math History
I'm trying to find good books on the history of mathematics, dating as far back as possible. There was a similar question here Good books on Philosophy of Mathematics , but mostly pertaining to Philosophy, and there were no good recommendations on…
Philoxopher
- 431
69
votes
11 answers
A linear operator commuting with all such operators is a scalar multiple of the identity.
The question is from Axler's " Linear Algebra Done Right ", which I'm using for self-study. We are given a linear operator $T$ over a finite dimensional vector space $V$. We have to show that $T$ is a scalar multiple of the identity iff $\forall S…
abeln
- 625
69
votes
4 answers
Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?
Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} \leqslant \sum_{n=1}^{\infty} \frac{1}{n^{1…
user55114
69
votes
5 answers
Help understanding Algebraic Geometry
I while ago I started reading Hartshorne's Algebraic Geometry and it almost immediately felt like I hit a brick wall. I have some experience with category theory and abstract algebra but not with algebraic or projective geometry. I'm wondering if…
user25470
- 1,083
- 2
- 10
- 15
69
votes
4 answers
Intersection of finite number of compact sets is compact?
Is the the intersection of a finite number of compact sets is compact? If not please give a counter example to demonstrate this is not true. I said that this is true because the intersection of finite number of compact sets are closed. Which…
John Buchta
- 701