Skip to main content
Search type Search syntax
Tags [tag]
Exact "words here"
Author user:1234
user:me (yours)
Score score:3 (3+)
score:0 (none)
Answers answers:3 (3+)
answers:0 (none)
isaccepted:yes
hasaccepted:no
inquestion:1234
Views views:250
Code code:"if (foo != bar)"
Sections title:apples
body:"apples oranges"
URL url:"*.example.com"
Saves in:saves
Status closed:yes
duplicate:no
migrated:no
wiki:no
Types is:question
is:answer
Exclude -[tag]
-apples
For more details on advanced search visit our help page
Results tagged with
Search options not deleted user 9025

eigenvalues of matrices or operators

2 votes

Quick tests to differentiate eigenvalues

When I had to do this with a couple of billion matrices, I computed the traces of some powers before going for the full test. A good method is to compute $\mathrm{tr} \,((A+xI)^{2^i})$ for $i=1,2,\ldo …
Brendan McKay's user avatar
6 votes
2 answers
251 views

Eigenvalues of polynomials of two matrices

In this question, the matrices are square and real and the polynomials have real coefficients, but feel free to mention other fields if that is interesting. Let $\chi(M)$ denote the characteristic pol …
Brendan McKay's user avatar
0 votes

An $n$ eigenvalue multiplicity

Let each $A_i$ be a matrix with all entries 0 except for the $(i,i+1)$ entry which is 1, where $i+1=1$ if $i=n$. The characteristic polynomial of $\sum_j a_j A_j$ is $x^n - \prod_j a_j$. I believe tha …
Brendan McKay's user avatar
13 votes

How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}( …
Brendan McKay's user avatar
2 votes
Accepted

Relation of row sums to largest eigenvalue

You are asking for relationships between the maximum independent set and the eigenvalues. If you search with those terms you will find several. … Haemers proved that the maximum size of an independent set is bounded above by $$ n\frac{-\lambda_1\lambda_n}{\delta^2-\lambda_1\lambda_n},$$ where $\lambda_1,\lambda_n$ are the largest and smallest eigenvalues
Brendan McKay's user avatar
16 votes
Accepted

What happens to eigenvalues when edges are removed?

The smallest eigenvalue can go up or down when an edge is removed. For "down": $G=K_n$ for $n\ge 3$. For "up": Take $K_n$ for $n\ge 1$ and append a new vertex attached to a single vertex of the orig …
Brendan McKay's user avatar