All Questions
Tagged with eigenvalues matrices
323 questions
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Summation of eigenvalues of tri-diagonal matrix smaller than specific value
Is there any analytic expression for summation of eigen-values of a tri-diagonal matrix which are smaller than a constant value? Or even a rough approximation for it. How about case of a general ...
- 1
0
votes
1
answer
139
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Spectrum of a Laplacianized matrix
Suppose that $A$ is a positive matrix and that we let $R$ be the diagonal matrix of $A$'s row-sums. What can be said about the spectrum of $R-A$? I am particularly interested in the largest eigenvalue ...
- 6,962
9
votes
1
answer
3k
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Connection between eigenvalues of matrix and its Laplacian.
Hello!
There are two definitions of graph spectrum:
1) Eigenvalues of adjacency matrix $A$.
2) Eigenvalues of Laplacian of adjacency matrix ($L$).
Different sources offer different properties based ...
- 89
10
votes
1
answer
1k
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Relationship between eigenvalues of $A-B$ and eigenvalues of $A^2-B^2$
Let us suppose that $A_{n}$ and $B_n$ are sequences of positive definite matrices satisfying
$$c \leq \lambda_{\min}(A_n)\leq \lambda_{\max}(A_n)\leq C$$
and
$$c \leq \lambda_{\min}(B_n)\leq \...
- 103
5
votes
1
answer
2k
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Condition for block symmetric real matrix eigenvalues to be real
I have a $2n \times 2n$ block symmetric matrix that in the simplest case ($n=2$) looks like:
$$
M_2 = \begin{bmatrix}
a_1 & 0 & b_{1,2} & -b_{1,2}\\\
0 & -a_1 & b_{1,2} & -b_{...
- 151
8
votes
1
answer
7k
views
Upper bound on largest eigenvalue of a real symmetric $n \times n$ matrix with all main diagonal entries positive, everywhere else nonpositive
Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
- 83
2
votes
0
answers
1k
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Eigenvalue problem for symmetric block tridiagonal matrices?
Is there a procedure to find the eigenvalues of $\textbf{M}$?
$$\begin{eqnarray}
\textbf{M}=\left[
\begin {array}{ccccc}
\textbf{A} & \textbf{B} & & &\\
\...
- 21
8
votes
2
answers
15k
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Upper bounds on eigenvalues of PSD matrix?
Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums?
For instance, the Perron–Frobenius ...
- 301
3
votes
0
answers
221
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Eigenvalues vs.matrix sparsity
For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity?
Is there a closed-form expression that states this ...
- 31
2
votes
1
answer
158
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Destroying the structure of a linear system while preserving its maximum eigenvalue
I have an asymmetric square matrix with non-negative real entries in the range [0,10], representing the edge-weights of a directed network. Assume that the network is a linear system. My general ...
- 21
36
votes
2
answers
32k
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Eigenvalues of the product of two symmetric matrices
This is mostly a reference request, as this must be well-known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
- 2,600
2
votes
1
answer
8k
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Properties of eigenvalues of general nonnegative matrices
I am aware, that an answer to this question can be found via Perron-Frobenius theory or something very similar, but unfortunately I am far from being an expert in the field and I am unable to find the ...
- 83
11
votes
3
answers
1k
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Maximum singular value of a random $\pm 1$ matrix
Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in ...
- 199
3
votes
1
answer
4k
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Eigenvalues of the sum of a positive matrix and a positive rank-$1$ matrix
Assume that $A, B$ are positive $n \times n$ matrices and that $B$ is rank-$1$, i.e., $B=xx^*$. If the eigenvalues of $A$ are $a_1 \geq a_2 \geq \cdots \geq a_n$, and $x$ is not an eigenvector of $A$, ...
- 31
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465
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Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits
Let $Sym^2(V)$ be the set of symmetric matrices of a real $n$-dimensional vector space $V$. Given an element $\underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n$, where $\lambda_1\leq\...
- 1,452
15
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4
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7k
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Eigenvectors and eigenvalues of a tridiagonal Toeplitz matrix
Is it possible to analytically evaluate the eigenvectors and eigenvalues of the following $n \times n$ tridiagonal matrix
$$
\mathcal{T}^{a}_n(p,q) = \begin{pmatrix}
0 & q & 0 & 0 &...
- 153
6
votes
0
answers
10k
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Eigenvalues of the sum of two matrices [duplicate]
I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B = C$ as a function of the eigenvalues of $A$ and of the eigenvalues of $B$ is generally a non-easy problem. I was ...
- 61
5
votes
2
answers
2k
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rank-one perturbation of a matrix corresponding to a specific spectrum
Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$.
Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
- 111
19
votes
1
answer
2k
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Smallest eigenvalue of a tricky random matrix
While experimenting with positive-definite functions, I was led to the following:
Let $n$ be a positive integer, and let $x_1,\ldots,x_n$ be sampled from a zero-mean, unit variance gaussian. Consider ...
- 28.6k
5
votes
2
answers
6k
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Computing the largest eigenvalue of a very large sparse matrix
I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter $w$, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue ...
- 233
21
votes
2
answers
3k
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Integer matrices with no integer eigenvalues
Let $$A = \begin{pmatrix} 3&1 \\ 0&1 \end{pmatrix}$$ and $$B = \begin{pmatrix} 1&0\\ 1&2 \end{pmatrix}$$ I want to show that the only elements of the semigroup generated by $A$ and $B$...
- 1,045
21
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5
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2k
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The middle eigenvalues of an undirected graph
Let $ \lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_{2n} $
be the collection of eigenvalues of an adjacency matrix of an undirected graph $G$ on $2n$ vertices. I am looking for any work or references ...
- 784
91
votes
5
answers
124k
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Eigenvalues of matrix sums
Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when the matrices are Hermitian and positive definite?
I am ...
- 1,013