All Questions
Tagged with matrices eigenvalues
323 questions
10
votes
0
answers
237
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 909
1
vote
0
answers
34
views
A recap of regularity of singular values as a function over M_n
So the core of the question is the study of the function $$ s :M_n(\mathbb R) \mapsto M_n(\mathbb R)$$
$$A \rightarrow s_n(A) $$
where $s_n(A)$ is the greatest singular value of A. I know there has ...
- 245
0
votes
1
answer
262
views
Perturbing a normal matrix
Let $N$ be a normal matrix.
Now I consider a perturbation of the matrix by another matrix $A.$
The perturbed matrix shall be called $M=N+A.$
Now assume there is a normalized vector $u$ such that $\...
user136156
3
votes
1
answer
463
views
Spectrum of this block matrix
Consider the following block matrix
$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$
where all submatrices are square and
matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
- 536
2
votes
0
answers
613
views
Smallest eigenvalue for Gram matrix of unit norm matrices
Given $n$ symmetric matrices $A_1, \dots, A_n \in \mathbb{R}^{k\times k}$, such that $\|A_i\| \leq 1$ for all $i$, we consider the matrix $M \in \mathbb{R}^{n\times n}$, where $M_{ij} = \langle A_i, ...
- 121
3
votes
2
answers
2k
views
Completely positive matrix with positive eigenvalue
A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$.
All eigenvalues of $A$ ...
- 133
1
vote
0
answers
111
views
Matrix eigenvalues inequality (2)
Suppose that $A$ is a $n\times n$ positive matrix, whose eigenvalues are $a_1\ge a_2\ldots \ge a_n>0;$ $B$ is a $m \times m$ positive matrix, whose eigenvalues are $b_1\ge b_2\ldots \ge b_m>0;$ ...
- 119
1
vote
0
answers
92
views
An inequality concerning the eigenvalues and eigenvectors of an SPD matrix
Let $Ax_i=\lambda_ix_i, \ (i=1,\cdots,n)$ be an eigensystem of the symmetric positive-definite diagonally-dominant matrix $A=\{a_{ij}\}$. Let
$$b_{jk}=\sum_{i=1}^{n}{\frac{(x_i(j)-x_i(k))^2}{\...
- 11
3
votes
1
answer
726
views
Largest eigenvalue of product of orthogonal-projection rank-1 perturbation
Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...
- 103
1
vote
0
answers
171
views
Eigenvalues of non-negative block matrices
$B$ is a non-negative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
- 21
5
votes
2
answers
249
views
Eigenvalue density of a symmetric tridiagonal matrix
Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
- 2,712
4
votes
1
answer
372
views
Eigenvalues of random matrix conditional on positive definiteness
Consider the Gaussian Orthogonal Ensemble, considered as a probability measure $\mu$ on the space of real symmetric matrices. Let $\mu|PD$ denote this measure conditioned on the event that the matrix ...
- 505
2
votes
1
answer
450
views
Eigenvalues of A^T D A for positive A and diagonal D
Suppose I have a diagonal matrix $D$ whose entries are bounded in absolute value. I also have a matrix $A$ that is positive (entry-wise, so $A_{ij} > 0\ \forall\ i,j$): one can assume that the ...
- 269
3
votes
2
answers
453
views
Find parameter values for which a 3x3 matrix has a triple eigenvalue
An Exceptional point generally occurs in eigenvalue problems in which the matrix is dependent on some parameter(s). The particular point in which the eigenvalues become degenerate for the parameter(s) ...
- 31
3
votes
1
answer
720
views
What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?
I have two problems related to eigenvalues of negative definite matrices:
I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[...
- 63
3
votes
0
answers
498
views
Eigenvectors of sum of SO(3) matrices
I asked this question before on MSE but go no answers. It seems that the problem is rather difficult so I thought of trying here. Given two matrices $A,B\in SO(n)$, each describing a rotation by ...
- 139
5
votes
1
answer
635
views
Largest Eigenvalue of a Matrix with Special Form in terms of n
In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern:
$$A_n = \begin{bmatrix}
0 & 0 & 0 & 0 &\dots & 0 \\
...
- 103
1
vote
0
answers
290
views
Upper bounds on absolute eigenvalue of sum of two matrix
We have this iteration
$$X_{k+1}=(G\cdot Jf+H)X_k+C$$
with $G$ is symmetric and nonnegative, $H$ is nonnegative. $Jf$ is the jacobian matrix of some function $f$ and we can assume it satisfy certain ...
- 153
4
votes
1
answer
205
views
How to find the analytical representation of eigenvalues of the matrix $G$?
I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
- 153
-2
votes
1
answer
967
views
What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]
Suppose we have the following symmetric matrix.
$$A = \sigma^2 I + u u^T$$
What can we say about the eigendecomposition of $A$?
- 67
1
vote
0
answers
359
views
Sufficient conditions for all eigenvalues simple in stochastic matrix
The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple.
...
- 11
1
vote
1
answer
506
views
Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues
For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset
$$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$
is a manifold of dimension $2n(2r)-(...
- 143
2
votes
1
answer
316
views
When does a row standardized adjacency matrix have a real spectrum?
A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
- 6,001
0
votes
0
answers
107
views
Numerical error on the spectrum of a matrix
Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
- 16.2k
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
1
answer
380
views
Rotatable matrix, its eigenvalues and eigenvectors
We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change.
I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
4
votes
0
answers
284
views
Maximizing a certain eigenvalue ratio
Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
- 2,712
2
votes
0
answers
330
views
Eigenvalues of special sum of Hermitian matrices
In my research on linear algebra and its applications, I have come across the following problem which has stumped me:
Let $ A $ be a positive definite matrix and let $ D $ be a positive diagonal ...
- 620
0
votes
1
answer
269
views
Limit of eigenvalues of a matrix perturbation sequence
Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...
- 135
7
votes
1
answer
641
views
Lower bound on the eigenvalues of the Laplacian
I am looking for a graph for which $2 d_{i} < \mu_{i}$, for some index $i$, where $\mu_{1} \leq \mu_{2} \leq \dots\leq \mu_{n}$ are the eigenvalues of the Laplacian matrix $L(G)$ and $d_{1} \leq d_{...
- 123
3
votes
1
answer
600
views
« Generalized simultaneous diagonalization » of a pair of symmetric, non-commuting, positive semi-definite matrices
I hope my question is trivial for some of you but for the time being I’m lost somewhere between the generalized eigenproblem, simultaneous diagonalization of quadratic forms, simultaneous SVD, ...
- 1,026
2
votes
1
answer
361
views
How the eigenvalues change when a Hermitian matrix is left multiplied and right multiplied by a diagonal matrix?
Suppose there is a Hermitian matrix $S$ with eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_K$. There is a diagonal matrix $D$ whose entries on the main diagonal are positive. What are the ...
- 21
0
votes
0
answers
224
views
Upper bound on matrix perturbation such that all eigenvalues lie within the unit circle
Consider the matrix
$$N=\left[\matrix{\mathbb{I}_n-\epsilon L & X\\ \epsilon Y & Z}\right]$$
where $\epsilon>0$ is a small positive parameter and $Z$ is a square $m\times m$ matrix with ...
- 101
5
votes
1
answer
8k
views
Eigenvalues and eigenvectors of tridiagonal matrices
What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...
- 73
2
votes
0
answers
550
views
Eigenvalues of a specific Hankel matrix
I have an $\frac{N}{2} \times \frac{N}{2}$ matrix $G$ with entries given by
\begin{equation}
G_{ij} = \frac{1}{\sin(\frac{\pi}{N}(i+j-\frac{3}{2}))}, \;\;\;\;\;\;\;\; 1 \le i,j \le \frac{N}{2},
\end{...
- 101
7
votes
3
answers
1k
views
Checking positive semi-definiteness of integer matrix
Key Problem : Is there any theorem about eigenvalues or positive semi-definiteness of small size matrices with small integer elements?
I have to check positive semi-definiteness of many symmetric ...
- 73
3
votes
1
answer
3k
views
Bounds for eigenvalues of block matrix
Let's say I have a block matrix of the form
$$X = \begin{bmatrix} A & B\\ B^T & C\end{bmatrix}$$
where $A$, $C$, and $X$ are all positive definite. I have bounds on both the minimum and ...
15
votes
1
answer
1k
views
Existence of double eigenvalue
Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent.
Must there exist a nonzero real linear combination $aA + bB$ which has a repeated ...
- 42.8k
1
vote
0
answers
19
views
Empirical approaches to validate observational bounds on minimum gap between least eigenvalues of $n \times n$ correlation matrix and its submatrices
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an $(n-1) \times (n-1)$ submatrix of $\Sigma$ obtained by eliminating the $i$-th row ...
- 141
7
votes
0
answers
264
views
Bound on gap between least eigenvalues of $n \times n$ correlation matrix and of its $(n -1) \times (n-1)$ submatrices
The following problem is motivated by one of my research problems.
Let
$\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$.
$\Sigma_i'$ be an ...
- 141
5
votes
3
answers
204
views
Find eigenvalues with given multiplicity in presence of errors
Let $A^0$ a $3 \times 3$ real symmetric matrix with eigenvalues $\lambda_1^0 = \lambda_2^0 \neq \lambda_3^0$. Hence, $\lambda_1^0$ has multiplicity $2$.
In real-world applications, I will have a ...
- 61
3
votes
2
answers
2k
views
Expected value of the largest singular value of a random matrix with entries in $N (0,1)$
Given a matrix $A \in \mathbb R^{n \times n}$ whose entries are i.i.d. $N(0,1)$, what is the expected value of its largest singular value? Equivalently, what is the expected value of the largest ...
- 33
12
votes
2
answers
1k
views
Eigenvalue perturbation theory via Feynman diagrams
Suppose I have a matrix given by a sum
$$A=D+\epsilon B$$
where $D$ is diagonal and $\epsilon$ is small, and I want the eigenvalues of $A$ as a power series in $\epsilon$. The first two orders in ...
- 1,549
2
votes
1
answer
799
views
Relation between LDLT and eigendecomposition of real symmetric matrices
The wikipedia page for Cholesky decomposition says:
For real matrices, the factorization has the form $A = LDL^T$ and is
often referred to as LDLT decomposition. It is
closely related to the ...
- 121
3
votes
1
answer
1k
views
The largest eigenvalue of a binary matrix with specific density
I would like to find the largest eigenvalue of an $n \times n$ binary matrix of density $p$, i.e., with $p n^{2}$ ones and $(1-p) n^{2}$ zeros. Any idea or reference is welcome.
- 159
7
votes
1
answer
3k
views
How can I calculate eigenvalues of a tridiagonal matrix? [closed]
Are there special methods to get exact eigenvalues of a tridiagonal matrix?
7
votes
0
answers
197
views
A special eigenvalue problem
For my research I need to solve a generalised eigenvalue problem
$Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
- 492
3
votes
1
answer
450
views
Finding non-symmetric matrix given real eigenvalues and eigenvectors [closed]
Given three eigenvectors and three eigenvalues, how would you go about finding BOTH non-symmetric matrix A and symmetric matrix B?
EDIT 7/27:
Sorry for not being specific enough~ In the problem I am ...
- 33
3
votes
0
answers
1k
views
Eigenvalues of block-hermitian matrices with zero diagonal blocks
I have a matrix of the form
$$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$
where $C$ is not necessarily hermitian. In general, can we say anything about the ...
- 47
6
votes
0
answers
138
views
A question on deformation theory of triples of matrices
Let $(x,y,z)$ be a triple of $n \times n$ traceless complex matrices which are simultaneously diagonalizable. We call such a triple regular if $C_x \cap C_y \cap C_z$ is a Cartan subalgebra of $\...
- 5,215