Articles of faq

Eigenvalues of the principal submatrix of a Hermitian matrix

This question aims at creating an “abstract duplicate” of various questions that can be reduced to the following: Let $A$ be an $n\times n$ Hermitian matrix and $B$ be an $r\times r$ principal submatrix of $A$. How are the eigenvalues of $A$ and $B$ related? Here are some questions on this site that can be […]

Find $6^{1000} \mod 23$

This question already has an answer here: How do I compute $a^b\,\bmod c$ by hand? 7 answers

Show that the set of all finite subsets of $\mathbb{N}$ is countable.

Show that the set of all finite subsets of $\mathbb{N}$ is countable. I’m not sure how to do this problem. I keep trying to think of an explicit formula for 1-1 correspondence like adding all the elements in each subset and sending that sum to itself in the natural numbers, but that wouldn’t be 1-1 […]

Single Variable Calculus Reference Recommendations

This question is a generalization of the common question asking for calculus references. It is here to abstract away the repetition, and give a canonical resource for calculus references. I’m looking for a resources to learn single-variable calculus.

Proof of upper-tail inequality for standard normal distribution

$X \sim \mathcal{N}(0,1)$, then to show that for $x > 0$, $$ \mathbb{P}(X>x) \leq \frac{\exp(-x^2/2)}{x \sqrt{2 \pi}} \>. $$

Differentiation of $x^{\sqrt{x}}$, how?

The answer is (I think) $x^{\sqrt{x}-0.5} (1+0.5\ln(x))$, but how?

Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix

This question aims to create an “abstract duplicate” of numerous questions that ask about determinants of specific matrices (I may have missed a few): Eigenvalues of a matrix of $1$'s Eigenvalues for the rank one matrix $uv^T$ Calculating $\det(A+I)$ for matrix $A$ defined by products How to calculate the following determinants (all ones, minus $I$) […]

Intuition behind Matrix Multiplication

If I multiply two numbers, say $3$ and $5$, I know it means add $3$ to itself $5$ times or add $5$ to itself $3$ times. But If I multiply two matrices, what does it mean ? I mean I can’t think it in terms of repetitive addition. What is the intuitive way of thinking […]

$\sqrt a$ is either an integer or an irrational number.

I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown that if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

Can a finite sum of square roots be an integer?

Can a sum of a finite number of square roots of integers be an integer? If yes can a sum of two square roots of integers be an integer? The square roots need to be irrational.