Most Popular
1500 questions
220
votes
8 answers
Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?
I'm trying to intuitively understand the difference between SVD and eigendecomposition. From my understanding, eigendecomposition seeks to describe a linear transformation as a sequence of three basic operations ( $P^{-1}DP$ ) on a vector: Rotation…
user541686
- 14,366
216
votes
9 answers
Importance of Representation Theory
Representation theory is a subject I want to like (it can be fun finding the representations of a group), but it's hard for me to see it as a subject that arises naturally or why it is important. I can think of two mathematical reasons for studying…
Eric O. Korman
- 20,165
216
votes
29 answers
What is a good complex analysis textbook, barring Ahlfors's?
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann…
MBP
- 1,235
215
votes
32 answers
Counterintuitive examples in probability
I want to teach a short course in probability, and I am looking for some counterintuitive examples in probability. I am mainly interested in the problems whose results seem to be obviously false while they are not. I already found some things. For…
MR_BD
- 6,477
214
votes
22 answers
Is it faster to count to the infinite going one by one or two by two?
A child asked me this question yesterday: Would it be faster to count to the infinite going one by one or two by two ? And I was split with two answers: In both case it will take an infinite time. Skipping half of the number should be really faster.…
Thomas Ayoub
- 1,645
213
votes
24 answers
Taking Seats on a Plane
This is a neat little problem that I was discussing today with my lab group out at lunch. Not particularly difficult but interesting implications nonetheless Imagine there are a 100 people in line to board a plane that seats 100. The first person in…
crasic
- 5,199
213
votes
4 answers
Limit of $L^p$ norm
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$ , $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$ ? I don't know where to start.
Parakee
- 3,504
213
votes
15 answers
Identification of a quadrilateral as a trapezoid, rectangle, or square
Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel ,…
The Chaz 2.0
- 10,666
212
votes
7 answers
How could we define the factorial of a matrix?
Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any sense, with $$\mathsf{A} = \left(\begin{matrix} 1 &…
user266764
211
votes
32 answers
Books on Number Theory for Layman
Books on Number Theory for anyone who loves Mathematics? (Beginner to Advanced & just for someone who has a basic grasp of math)
Prasoon Saurav
- 610
211
votes
2 answers
How to show $e^{e^{e^{79}}}$ is not an integer
In this question , I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? After several years, it appears this is an open problem. As a non-number…
Carl Mummert
- 87,231
210
votes
1 answer
Are $14$ and $21$ the only "interesting" numbers?
The numbers $14$ and $21$ are quite interesting. The prime factorisation of $14$ is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$ . Note that $3$ is the prime after $2$ and $5$ is the prime before $7$ . Similarly, the prime…
Simon Parker
- 4,431
209
votes
8 answers
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression?
How can we sum up $\sin$ and $\cos$ series when the angles are in arithmetic progression? For example here is the sum of $\cos$ series: $$\sum_{k=0}^{n-1}\cos (a+k \cdot d) =\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times \cos \biggl(…
Quixotic
- 22,915
209
votes
7 answers
Do we have negative prime numbers?
Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
user103028
208
votes
91 answers
Which one result in mathematics has surprised you the most?
A large part of my fascination in mathematics is because of some very surprising results that I have seen there. I remember one I found very hard to swallow when I first encountered it, was what is known as the Banach Tarski Paradox . It states that…
KalEl
- 3,437