All Questions
Tagged with matrices eigenvalues
323 questions
17
votes
3
answers
2k
views
Finding the nearest matrix with real eigenvalues
In this thread on MATLAB Central, I found a discussion on finding the nearest matrix with real eigenvalues. The first hypothesis was to simply truncate the complex part of the eigenvalues. So, given ...
- 185
1
vote
0
answers
650
views
Bound of the eigenvalues of a matrix product of two diagonal and one symmetric PSD matrices
Let B be a positive diagonal matrix, and C a PSD symmetric matrix with eigenvalues $\Lambda$, with $\lambda_i = {0,1}$. I'm trying to find a reference for the following bounds or similar
$B^{1\over 2}...
- 111
7
votes
1
answer
2k
views
Bound the eigenvalue of product of matrices?
Let $H$ be a $n \times n$ real symmetric matrix that has eigenvalues with absolute value less than 1. Define the matrix $M = \prod_{i=1}^n (I - e_ie_i^{\top}H)$ where $e_i$ denotes the $i^\text{th}$ ...
user31317
4
votes
0
answers
2k
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What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
- 123
3
votes
0
answers
220
views
Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
2
votes
1
answer
1k
views
Is the sum of two stable matrices also stable?
Let $A$ and $B$ be two arbitrary real matrices of the same dimension. If the eigenvalues of $A$ and $B$ are all in the left half of the complex plane, can we estimate the the location of the ...
- 23
1
vote
0
answers
129
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Matrix majorization when a diagonal matrix is multiplied from right and left
Let $D_1$ and $D_2$ be two diagonal matrices such that $D_1^2+D_2^2=I$ (identity matrix). Suppose matrix $A$ majorizes matrix $B$. Can we show that matrix $A$ majorizes matrix $D_1 A D_1 + D_2 B D_2$? ...
- 181
2
votes
1
answer
728
views
Maximum eigenvalue of Hadamard power of a positive semidefinite matrix
Let $K$ be a covariance matrix. It is positive semidefinite, its diagonal elements are all 1, and its off-diagonals are between -1 and 1. Let $K.^2$ be its element-wise power (Hadamard power). Can we ...
- 181
5
votes
3
answers
2k
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Proving that a certain non-symmetric matrix has an eigenvalue with positive real part
Suppose that
$X$ is the $n \times n$ matrix of all ones
$Y$ is an arbitrary $n \times n$ matrix with zeroes on the diagonal and all other entries equal to $0$ or $1$
$0 < \delta < 1$
Let $Z = ...
- 61
1
vote
2
answers
523
views
SOLVED: Multiplicity of the eigenvalues of the sum of two matrices [closed]
* Question Solved *
This question ultimately was about the conditions for violation of the Wigner-von Neumann non-crossing rule, which is still an open field of research.
Thank you very much to all ...
- 41
2
votes
0
answers
148
views
Commutation relation and eigenvectors of infinite matrices [closed]
I'm given the Matrix $A$ and $A^T$:
$A =
\begin{bmatrix}
0 & 1 & 0 & 0 & \dots \\
0 & 0 & \sqrt{2} & 0 & \dots \\
0 & 0 & 0 & \sqrt{3} & \...
- 121
8
votes
1
answer
904
views
A generalized log inequality for positive definite trace-one matrices
Let $\{V_i\}_{i=1}^N$ be a set of $n\times m$, $n\geq m$, real matrices of full column rank and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. Moreover, let $A^{1/2}=(...
- 2,712
12
votes
3
answers
737
views
How to find Suleimanova's work on the Nonnegative Inverse Eigenvalue Problem?
Many papers cite the work of Suleimanova when studying inverse eigenvalue problems - in particular, the nonnegative inverse eigenvalue problem (NIEP). However, I cannot seem to find her work anywhere....
- 121
9
votes
0
answers
624
views
Eigenvalues of leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix
It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix
$$\begin{pmatrix}
0 & n-1 & 0 & \dots & 0 \\\
1 & 0 & n-2 & \dots & 0\\\
0 & ...
- 139
14
votes
5
answers
989
views
Eigenvalues of a matrix with entries involving combinatorics
Let $F(n, l, i, j)$ be the cardinality of the set
\begin{eqnarray*}
\{(k_1, \cdots, k_n)\in\mathbb{Z}^{\oplus n}|0\leq k_r\leq l-1\text{ for }1\leq r\leq n\text{, }k_1+\cdots+k_n=lj-i\}.
\end{eqnarray*...
- 383
6
votes
1
answer
879
views
A question on the smallest singular value
Let $X(r)$ be the set of matrices $A \in M(n \times m)$, $n \leq m$, such that the norm of $A$ (largest singular value) is smaller or equal than $1$ and the smallest singular value of $A$ is smaller ...
- 61
4
votes
0
answers
84
views
Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
9
votes
3
answers
383
views
convergence of 2nd eigenvalue
Fix $0<h_1<h_2<h_3<1$ reals. All matrices below are $3\times3$ real.
Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric ...
- 43.2k
5
votes
1
answer
375
views
Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?
I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational ...
- 1
18
votes
1
answer
847
views
Showing that a certain matrix is not positive definite
Let $J_k$ be a $k \times k$ all ones matrix and $B$ any $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$.
I would like to show that the matrix $$X_B = (J_k -I) - B (J_k - I)^{-...
- 3,463
1
vote
0
answers
398
views
Center of matrices
I encountered a neat problem in a problem in particle physics
So given $n$ skew symmetric matrices $A_1,...,A_n$ in $\mathbb{C}^{d \times d}.$
I would like to call this the commutator property: $...
1
vote
0
answers
264
views
Maximizing the ratio of largest eigenvalues
Let $K$ be a real stable matrix; more specifically,
$$
K=\left(\begin{array}{rrrrr}
0&1&0&\ldots&0\\
0&0&1&\ldots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\...
- 270
4
votes
1
answer
545
views
No arbitrary product of matrices has eigenvalue 1?
Consider the matrix $D$, adjacency matrix of an undirected graph $G$ on $n$ vertices, with the notation that $d_{i,i}=0,\forall i$.
The matrices $A_i$ are constructed from Identity matrices, $I_{n*n}$...
- 143
-2
votes
1
answer
432
views
Eigenvalues of cyclic tridiagonal matrix [closed]
The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum
n_i=n.$ Prove that this matrix ...
- 640
2
votes
1
answer
483
views
Modified interlacing of eigenvalues
Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A &...
- 135
2
votes
4
answers
293
views
Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle
...or prove that none exists.
Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory).
Preferably the ...
- 21
1
vote
1
answer
2k
views
Largest element in inverse of a positive definite symmetric matrix [closed]
If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
- 15
1
vote
1
answer
307
views
Can we claim that all the terms in a matrix are less than equal to 1 if spectral radius is less than 1?
I have a a full column rank matrix A, and using this I want to construct a matrix with spectral radius less than 1. I do that using,
H = $I-\alpha A^{T} A$ ($I$ is identity matrix), where the term $\...
- 15
11
votes
1
answer
980
views
Exact eigenvalues of a specific tridiagonal matrix
I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
- 161
0
votes
1
answer
106
views
Eigenvalue-related statements [closed]
(I understand this question might not be appropriate for this website, but it has been asked on MathStackexchange and did not receive any replies even with a bounty)
How can I prove that the ...
- 103
13
votes
2
answers
1k
views
A log inequality for positive definite trace-one matrices
Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following ...
- 2,712
4
votes
1
answer
506
views
Eigenvalues of large tridiagonal matrix
Consider large tridiagonal matrix (where $a$ and $b$ are real numbers):
$$M =
\begin{pmatrix}
a^2 & b & 0 & 0 & \cdots \\
b & (a+1)^2 & b & 0 & \cdots & \\
...
- 285
3
votes
2
answers
290
views
Eigenspace of convex combination of two idempotent matrices
Let $H_1,H_2\in\mathbb{Q}^{n\times n}$ be idempotent and symmetric matrices. For any $0<\mu<\frac{1}{2}$, consider the matrix
$$H_\mu:=\mu H_1+(1-\mu)H_2.$$
I'm looking for a description of $\...
- 268
3
votes
1
answer
371
views
Eigenvectors of a perturbed reducible stochastic matrix
Let $Q$ be a $n\times n$ reducible stochastic matrix. Let $J$ be such that $[J]_{ij}={1 \over n}$. Now for a small positive constant $\alpha\in [0,1]$, consider the matrix
$$\tilde{Q}\,=\,(1-\alpha)...
- 1,421
8
votes
2
answers
12k
views
Relation between eigenvalues of $A$ and $A^TA$?
For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...
- 181
2
votes
1
answer
351
views
Eigenvalues of product of symplectic matrices
I have 2 symplectic matrices $X_{1},X_{2} \in \mathbb{R}^{2n\times2n}$. The matrix $X=X_{1} \cdot X_{2}$ is also symplectic.
Question: Are there any theorems which allow me to express eigenvalues of ...
- 133
7
votes
1
answer
412
views
Sum of the absolute eigenvalues of A>=B
Kindly help me to prove/disprove the following statement.
Let $A$ be a symmetric matrix of order $n \times n$ with all the diagonal entry equal to $0$, and other non-diagonal entry equal to $k$ (...
3
votes
3
answers
357
views
Can a block matrix with at least 3 zero blocks of different size on the diagonal and 1's everywhere else have only integer eigenvalues?
Let $M=\begin{pmatrix}
\begin{array}{cccccccc}
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
0 & 0 & 1 & 1 & 1 & 1 & 1 &1\\
1 & 1 & 0 & 0 & ...
4
votes
2
answers
477
views
Non-asympototic version of Gelfand's formula
Let $A$ be a $n\times n$ matrix. Let $\|A\|$ be the spectral norm of $A$, and $\rho(A)$ be the spectral radius. I am wondering whether the following statement is true.
There exists universal ...
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-1
votes
1
answer
230
views
How bad could $\|A^k\|$ be when $\rho(A) < 1-\delta$ [closed]
(Sorry, I do hate editing this many many times but let me try the last time)
Gelfand's formula says that
$$\lim_{k\rightarrow \infty} \|A^k\|^{1/k} = \rho(A)$$
I am wondering whether there is any ...
- 181
3
votes
0
answers
182
views
Relating Numerical Range and Perron-frobenius theorem for positive matrices?
Let $A$ be any matrix with all entries positive (which means Perron-Frobenius theorem can be applied). Then its numerical range is defined as the set of complex numbers
$$W(A)=\{x^HAx\lvert ~x^Hx=1\}$$...
- 1,421
-4
votes
1
answer
387
views
Eigenvalues of real symmetric matrix [closed]
Suppose $A$ is a $n \times n$ real symmetric matrix with entries $a_{ij}\geq 1 $ and $a_{ii} = 0 $. Is it possible to have sum of the absolute eigenvalues of
$A < 2 (n - 1).$
1
vote
1
answer
2k
views
Upper bound for sum of absolute values of eigenvalues of Hermitian matrix
Given a hermitian, but not necessarily positive, sparse matrix $C = (c_{ij}) \in \mathbb{C}^{n \times n}$ and $n \ggg 1$ ($n \approx 2^{100}$) with eigenvalues $\lambda_1 \le \lambda_2 \le \dots \le \...
- 13
0
votes
2
answers
588
views
Estimating the shift in the $\lambda_{\max}$ of a matrix under a diagonal perturbation
Given a matrix $A$ and a diagonal matrix $D$, how can we estimate $\lambda_{\max}(A+D) - \lambda_{\max}(A)$? Feel free to make other assumptions about the matrices that they are all symmetric and have ...
1
vote
1
answer
2k
views
Limit of largest eigenvalue [closed]
For positive definite matrix, if we increase the dimension to the infinity, is it true that the largest eigenvalue stays bounded from above?
In other words does the following limit exists:
$$\lim_{n\...
- 267
4
votes
1
answer
4k
views
Eigenvalues of the sum of two matrices, where one is $B=\operatorname{diag}(1, 0,\dots,0)$
I know that given two matrices $A$ and $B$, estimating the eigenvalues of $A + B$ by the eigenvalues of $A$ and $B$ is generally a non-easy problem. In particular, there are some results for matrices ...
- 297
8
votes
0
answers
2k
views
Possible values of eigenvalues of Hadamard product of Hermitian matrices
One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem:
...
- 891
3
votes
1
answer
559
views
Determinant of a Certain Positive-Definite Block Matrix
Is there a lower bound for the determinant or minimum eigenvalue of the following $d$ by $d$ matrix in terms of $d$?
$$\Gamma=\left( {\begin{array}{cc}
I & B \\
B^{*} & I \\
\end{array} ...
- 31
1
vote
0
answers
148
views
Perturbation of eigenvalues of some special matrices
In perturbation theory of linear operators, one major question is how the eigenvalues of a linear operator $A$ change under a small perturbation, $A(x) = A + xP$, with $x\in\mathbb{R}$. For instance, ...
- 61
7
votes
1
answer
6k
views
Eigenvectors as continuous functions of matrix - diagonal perturbations
The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
- 2,222