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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 …
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 …
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 …
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 $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}( …
2
votes
Accepted
Relation of row sums to largest eigenvalue
You are asking for relationships between the maximum independent set and the eigenvalues. If you search with those terms you will find several. … Haemers proved that the maximum size of an independent set is bounded above by $$ n\frac{-\lambda_1\lambda_n}{\delta^2-\lambda_1\lambda_n},$$ where $\lambda_1,\lambda_n$ are the largest and smallest eigenvalues …
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 …