Intereting Posts

Strange set notation (a set as a power of 2)?
$\sum_1^n 2\sqrt{n} – \sqrt{n-1} – \sqrt{n+1} $ converge or not?
$f(z_1 z_2) = f(z_1) f(z_2)$ for $z_1,z_2\in \mathbb{C}$ then $f(z) = z^k$ for some $k$
Question regarding an inequality
Eigenvalues for matrices over general rings
Number of subsets when each pair of distinct elements is contained in exactly one subset
Equivalence of measures and $L^1$ functions
There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$.
Probability problem
proving that this ideal is radical or the generator is irreducible
Expanding problem solving skill
Non-associative version of a group satisfying these identities: $(xy)y^{-1}=y^{-1}(yx)=x$
Find $g$ using euclids algorithm
Calculate the Wronskian of $f(t)=t|t|$ and $g(t)=t^2$ on the following intervals: $(0,+\infty)$, $(-\infty, 0)$ and $0$?
In NBG set theory how could you state the axiom of limitation of size in first-order logic?

The principal eigenvector of the adjacency matrix of a graph gives us some notion of vertex centrality.

What do the second, third, etc. eigenvectors tell us?

**Motivation:** A standard information retrieval technique (LSI) uses a truncated SVD as a low-rank approximation of a matrix. If we truncate to rank 1, then we essentially have a PageRank algorithm. I was wondering if there are ways of interpreting the corrections introduced by higher eigenvectors.

- If two real symmetric square matrices commute then does they have a common eigenvector ?
- Eigenvalue Problem — prove eigenvalue for $A^2 + I$
- The relationship between eigenvalues of matrices $XY$ and $YX$
- What is the relation between rank of a matrix, its eigenvalues and eigenvectors
- generalized eigenvector for 3x3 matrix with 1 eigenvalue, 2 eigenvectors
- If $A$ is any matrix then $A^*A$ and $AA^*$ are Hermitian with non-negative eigenvalues

Something similar to what the moments of a distribution tell us (e.g. first moment gives us the mean, second tells us the variance, third gives us skewness, etc).

- How to get the adjacency matrix of the dual of $G$ without pen and paper?
- Let $B$ be a nilpotent $n\times n$ matrix with complex entries let $A = B-I$ then find $\det(A)$
- Prove a graph with $2n-2$ edges has two cycles of equal length
- Characterizing a real symmetric matrix $A$ as $A = XX^T - YY^T$
- connection between graphs and the eigenvectors of their matrix representation
- Matrix Inverses and Eigenvalues
- Prove a graph Containing $2k$ odd vertices contains $k$ distinct trails
- Prove that if the sum of each row of $A$ equals $s$, then $s$ is an eigenvalue of $A$.
- Proof that any simple connected graph has at least 2 non-cut vertices.
- Average Degree of a Random Geometric Graph

The second eigenvalue of a markov chain has meaning, and it affects (for example) the convergence and stability of algorithms that find the equilibrium distribution. See The second eigenvalue of the Google matrix, by Haveliwala and Kamvar for a discussion on how to compute its value.

The second eigenvector, however, has to be something more complex. Given that the PageRank matrix is a reversible markov chain (by construction), it has a unique equilibrium distribution. So the only vector v for which P v = v is the first eigenvector. So, the second eigenvector does not necessarily represent a probability distribution. According to Fluctuation Induced Almost Invariant Sets, by Schwartz and Billings, if a reversible markov chain represents a linear dynamical system the second eigenvector is a good way of finding almost invariant sets, which are subsets of the pages in which the perfect stochastic user would spend a long time “trapped in” before leaving to other parts of the web. The signs of the entries of the second eigenvector (and the third, and maybe others) can be used to find these almost invariant sets.

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- Let $M_1,M_2,M_3,M_4$ be the suprema of $|f|$ on the edges of a square. Show that $|f(0)|\le \sqrt{M_1M_2M_3M_4}$
- Evaluate $\sum_{n=1}^\infty \frac{n}{2^n}$.
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- Generate Correlated Normal Random Variables