All Questions
Tagged with matrices eigenvalues
323 questions
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How to analyse the range of eigenvalues of a symmetric and indefinite matrix?
Let $G$ be a symmetric and indefinite matrix defined as follows
$$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$
where $S$ is a symmetric positive definite matrix of size $...
- 1
4
votes
0
answers
989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
15
votes
1
answer
474
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Has Nambu's notion of an "eigenoperator" found a place in the mathematical literature?
The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a ...
- 188k
2
votes
0
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69
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Unimodular eigenvalue of a H-self-adjoint matrix (indefinite innerproduct)
Let $A,H \in \mathbb{C}^{n \times n}$ be such that $H$ is Hermitian and invertible and $A = H^{-1} A^* H$. In this case, $A$ is said to be $H$-self-adjoint. This is due to the fact that if $\langle \...
- 175
2
votes
1
answer
142
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Spectral majorization for symmetric matrices
In ${\mathbb R}^n$, a vector $a=(a_1,\ldots,a_n)$ is said to majorize another vector $b=(b_1,\ldots,b_n)$ if for any convex function $f\colon\mathbb R\to\mathbb R$, we have
$$\sum_{i=1}^nf(a_i)\ge \...
- 234
3
votes
0
answers
538
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Diagonalizing a block tridiagonal matrix
Is there an efficient way to diagonalize a block tridiagonal $N\times N$ matrix of the following form:
\begin{matrix}
A_0 & B & 0 & 0 & \ldots \\
B & A_1 & B & 0 & \...
- 31
1
vote
1
answer
209
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Eigenvalues invariant under 90° rotation
Consider $N \times N$ matrices
$$A = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & & 0 \\
\vdots & 1 & 0 & \...
- 536
1
vote
0
answers
356
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Is there relation between eigenvalues of A,B,A+B,AB? [closed]
I found there are some works about eigenvalues of $A,B,A+B$ in the case of Hermitian matrices. For example, here.
I am wondering in the case that $A$ and $B$ is not Hermitian.
For example, my question ...
- 405
2
votes
1
answer
512
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Submatrices of matrices in $\mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ [closed]
This is a follow-up question to my question from Math Stackexchange (Thank you Dietrich Burde and Michael Burr for the help).
Let $M\in \mathrm{SL}(4, \mathbb{Z})$ with all eigenvalues equal to $1$ (i....
- 823
12
votes
1
answer
1k
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Eigenvalues come in pairs
Consider the two matrices with some parameter $s \in \mathbb R$
$$A_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$
and
$$...
- 124
2
votes
1
answer
244
views
Expected minimal distance of eigenvalues
Let $A$ be an arbitrary symmetric matrix and $B$ be a random GUE matrix. I would like to know. Are there any results on the minimal eigenvalue distance between two distinct eigenvalues of $A+B$? I ...
- 73
4
votes
1
answer
477
views
Are eigenvalues preserved under derived equivalence?
Let $A$ and $B$ be finite dimensional algebras such that $A$ and $B$ are derived equivalent.
Denote by $C_A$ (resp. $C_B$) the Cartan matrix of $A$ (resp. $B$).
Then does the set of eigenvalues of $...
- 49
2
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0
answers
537
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Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices
recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
- 21
2
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0
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121
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Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 21
3
votes
2
answers
853
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Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration
I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ ...
- 3,098
16
votes
3
answers
2k
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Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
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-2
votes
1
answer
261
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Proving 2 matrices have the same trace [closed]
I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is:
Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...
- 331
2
votes
1
answer
998
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Diagonalizing a symmetric block matrix
Let us consider the matrix
$$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$
then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$
Now, let us consider a block matrix
$$ A = \begin{...
- 29
5
votes
2
answers
343
views
Maximal eigenvalue of a correlation matrix with some entries fixed as zeros
Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
- 51
1
vote
1
answer
241
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Monotonicity of eigenvalues II
In a previous question here, I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-...
- 536
6
votes
1
answer
601
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Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 536
5
votes
1
answer
417
views
Log determinant of quadratic form
I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...
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6
votes
4
answers
1k
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Under what conditions are the eigenvalues of a product of two real symmetric matrices real?
Under what conditions are the eigenvalues of a product $M = A B$ of two real symmetric matrices $A$ and $B$ real?
And is there a way to relate the signs of the eigenvalues of $M$ to any properties of $...
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0
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46
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What kind of bounds for $\mathrm{Re}(\lambda(A))$ when $\lambda_{\mathrm{max}}(A + A^t) < 0$?
What can be said about the real parts of eigenvalues of $A \in \mathbb R^{n\times n}$ when $\lambda_{\max}(A + A^t) < 0$?
I think the real parts of eigenvalues of $A$ will be negative, but I can't ...
- 1
0
votes
1
answer
263
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Change in the largest eigenvalue due to perturbation of diagonal components of a symmetric matrix
Let $A\in \mathbb{R^{n\times n}}$ be a symmetric negative difinite matrix and
$D\in \mathbb{R}^{n\times n}$ be a diagonal matrix $D = \mathrm{diag}\{d_i\}, (d_i < 0)$.
From Weyl's inequality, the ...
- 1
2
votes
1
answer
74
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Limitation through the singular values
Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
- 143
6
votes
0
answers
396
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Typical eigenspectrum of a random projection of a large matrix
Suppose I have a real symmetric $m \times m$ matrix $\Lambda$. This matrix is large ($m \gg 1$) and, for simplicity, we'll assume it's diagonal. I then construct a random $n \times n$ projection
$$ A =...
- 453
2
votes
0
answers
345
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Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
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1
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0
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618
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Largest eigenvalue of matrix A smaller than 1, what about B when A=B+C? [closed]
Suppose I have a square matrix $A$ that only has non-negative real entries and is not symmetric and not primitive either. It has no "special" structure we could exploit. I know that the ...
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12
votes
4
answers
904
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Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts
Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix
...
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1
vote
1
answer
467
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Trace minimization for generalized eigenvalue problem
In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have
$$
\min_{Y \in Y^*} \text{tr}(Y^TAY) =
\text{tr}(X^TAX) =
\sum_{i=1}^p \lambda_i,
$$
with
$$
\text{
$X^...
- 21
1
vote
0
answers
331
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Eigenvalues of an (almost) pentadiagonal symmetric Toeplitz matrix
I am looking for analytic expressions for the eigenvalues of matrices of the form
$$A = \begin{bmatrix}
6 & -4 & 1 & 0 & 0 & 0 & 0 \\
-4 & 6 & -4 & 1 & 0 &...
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1
vote
1
answer
223
views
Growth rates of ‘sub-traces’ of matrices
Consider an aperiodic, non-negative and non-zero $m\times m$ matrix $A$. Here aperiodic means that there exists an integer $n$ such that every entry of $A^n$ is strictly positive. By the Perron–...
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2
votes
1
answer
156
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Minimal Laplacian spread of a graph
Laplacian spread of a graph is the difference among the largest and the second smallest Laplacian eigenvalue of the graph. Is there any result or conjecture that discusses about the graphs having ...
8
votes
0
answers
392
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Bounding eigenvalues by taking high powers of matrices: history?
Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that
$$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$
for any $k\geq 1$. ...
- 20.2k
2
votes
0
answers
81
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Perturbed Gram matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first canonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(x_t ...
- 215
2
votes
1
answer
392
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Eigenvalue perturbation under sparse perturbations
Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...
- 21
8
votes
2
answers
380
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Projecting onto space of matrices with spectral radius less than one
Consider the space
$$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$
where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times ...
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1
vote
1
answer
1k
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Prove that absolute value of eigenvalue is smaller than 1 [closed]
I want to prove that the absolute value of the eigenvalues of a matrix A are smaller than 1 for $$A=\left(\begin{array}{cc}
0 & -H_{11}^{-1} H_{12} \\
-H_{22}^{-1} H_{21} & 0
\end{array}\right)...
1
vote
1
answer
260
views
How can I obtain the eigenvalues of this matrix?
Consider the following $M \times 3$ matrix
$$\mathbf F = [\mathbf h_1, \mathbf h_2, \mathbf h_3],$$
with distinct non-zero singular values $\sigma_1 >\sigma_2 > \sigma_3$, where $\mathbf h_k$'s ...
- 31
2
votes
1
answer
236
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How can I prove a randomly generated matrix has distinct non-zero eigenvalues?
Consider the following $M×M$ matrix
$$
\mathbf A=\sum_{k=1}^K =a_k \mathbf h_k \mathbf h_k^H,(M≥K)
$$
where $a_k$'s are real values and $h_k$'s are $M×1$ randomly generated vectors, e.g., complex ...
- 31
11
votes
1
answer
927
views
Imaginary eigenvalues
Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is ...
- 124
2
votes
1
answer
212
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Similarity of two matrices
Consider the matrix, for some $\lambda \in \mathbb R$ .
$$A=\begin{pmatrix} i \lambda & -1 & i & 0 \\ 1 & 0 & 0& 0 \\ i & 0 & - i \lambda & -1 \\ 0 & 0 & 1 ...
- 124
13
votes
3
answers
2k
views
Eigenvalue pattern
We consider a matrix
$$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily ...
- 133
3
votes
1
answer
2k
views
Eigenvalues of a block matrix with zero diagonal blocks
Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix
\begin{equation}
M:=\begin{pmatrix}
0_{k_1} & A\\ A^\top & 0_{k_2}
\end{pmatrix},
\end{equation}
...
- 53
10
votes
2
answers
614
views
Lower eigenvectors of nonnegative matrices with zero trace
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, \...
- 1,897
1
vote
1
answer
125
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 ...
- 205
1
vote
0
answers
373
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 ...
- 245
1
vote
2
answers
565
views
How to find bounds on the eigenvalues of a matrix?
Given this matrix
$M=\begin{bmatrix} 0 & m-1 & 2 & n-1\\
1 & m-2& 1 & n-1\\
2 & m-1 & 0 & 2(n-1)\\
1 & m-1 & 2 & 2(n-2)\end{bmatrix},$ show that if $\...
- 444
2
votes
2
answers
201
views
Eigen problem with constrained (equal) eigenvalues
Let $\Omega$ be a symmetric and positive definite matrix. From a test of hypothesis I know that some eigenvalues are likely to be equal (the test also suggests which eigenvalues). Do you have any ...
- 131