# Questions tagged [eigenvalues]

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### Methods to find the spectrum of an operator

Suppose we have a bounded, self-adjoint operator $T$ on a set of functions $\mathcal{F}$. What kinds of methods are there to find the spectrum of $T$? Here is the setting I'm wondering about: consider ...
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### Proof (or reference) about $\lambda_i(A+\epsilon e_je_j^*) = \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2).$

I'm looking for a proof (or a reference in a textbook) about the fact that $$\lambda_i(A+\epsilon e_je_j^*) =_{\epsilon \to 0} \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2),$$ where $A$ is a ...
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### Probability finite precision random matrix has distinct eigenvalues

copied from math stack exchange There is a theorem which says the probability/size of a random matrix having repeated eigenvalues is 0 and this result is used in many fields. What I am wondering is, ...
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### Expansion in hypergraphs

Is there a useful concept of expansion in hypergraphs, generalizing the concept for graphs (see: expander graphs)? Of course, expander graphs can be characterized in several qualitatively equivalent ...
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Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \... 0answers 88 views ### Eigenvalues of splitting scheme In numerical analysis it is common to approximate a solution to a PDE $$u'(t) = (A+B) u(t), \quad u(0)=u_0$$ which is just given by$e^{t(A+B)}u_0$by the splitting$e^{tB/2} e^{tA} e^{tB/2}u_0.$Here,... 2answers 551 views ### On a matrix problem in the field$\mathbb F_2$Given$M$a symmetric matrix in$\mathbb F_2^{n\times n}$having$\mathsf{det}_\mathbb R(M)\neq0$(non-singular in reals) and satisfying$PMP'=(M+J+I)$or$P(M+J+I)P'=M$where$P$is a permutation ... 0answers 30 views ### Eigenvalue bounds of a random graph with a clique I'm looking into this paper and having some problems proving (ii) of proposition 2.1. I don't quite understand how the lemma is proved. I also read the original paper where the lemma comes from but ... 1answer 44 views ### A monotonicity property of eigenvalues Let$A \in S^{n}_{+}$be a positive semi-definite matrix and$D \in S^{n}_{+}$a diagonal matrix with all the diagonal entries no smaller than one, i.e.,$D_{ii} \geq 1$for all$i \leq n$. I wonder ... 0answers 104 views ### Upper bound on the sum of the smallest non-zero eigenvalues Let$\mathcal A := \{ A_1, A_2, \dots, A_n \} \subset \Bbb R^{d \times d}$be a set of symmetric and positive semidefinite matrices. For a matrix$A_k \in \mathcal A$, denote its (real) eigenvalues by ... 0answers 35 views ### Formulas to determine the value of graph energy with addition or deletion of edges If$G$is a graph, then the graph energy of$G$denoted by$E(G)$is defined as the sum of absolute values of eigenvalues of the adjacency matrix of$G$. It is known that$E(G)\geq E(G-v)$, where$ ...
I am working on an optimization problem (for example, conjugate gradient) to solve $Ax=b$, where $A$ is a symmetric positive definite matrix. I can understand that the CG (conjugate gradient) has ...