All Questions
202 questions
0
votes
0
answers
46
views
max eigenvalue and schatten-1 norm of depolarizing channel acting on trace-0 Hermitian matrix
Denote $\mathcal{H}^n$ as the $n-$dimension Hermitian matrices. Depolarizing channel $\Delta_p:\mathcal{H}^2\to\mathcal{H}^2$ is defined as $\Delta_p(A)=p\text{ tr }(A)~I/2+(1-p)A$ where $A\in \...
- 91
2
votes
1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 91
1
vote
0
answers
63
views
Reference request for non-banded Toeplitz matrix
I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix.
I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
- 11
0
votes
1
answer
309
views
Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
- 13
1
vote
1
answer
48
views
Iteration matrix representation with complex conjugate operator
I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
5
votes
2
answers
721
views
Matrices with same eigenvalues
This question is a more precise version of this question.
Let's assume we have the matrix
$$\left(
\begin{array}{ccccc}
0 & a & 0 & 0 & 0 \\
f & 0 & b & 0 & 0 \\
0 &...
16
votes
3
answers
2k
views
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 ...
- 73
2
votes
1
answer
512
views
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
91
votes
5
answers
124k
views
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
2
votes
1
answer
400
views
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
12
votes
1
answer
1k
views
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
$$...
- 124
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 4,058
2
votes
1
answer
299
views
Eigenvalues of a specific matrix
I have a block matrix
$$M=\begin{bmatrix}
I_0& I_1& \cdots& I_1\\
I_2& I_0& \ddots& \vdots\\
\vdots& \ddots& \...
- 43
13
votes
3
answers
2k
views
Eigenvalue pattern
We consider a matrix
$$M_{\mu} = \begin{pmatrix} 1 & \mu & 1 & 0 \\ -\mu & 1 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 &-1 & 0 & 0 \end{pmatrix}$$
One easily ...
- 133
16
votes
2
answers
1k
views
Spectral symmetry of a certain structured matrix
I have a matrix
$$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$
As ...
- 536
6
votes
4
answers
1k
views
Under what conditions are the eigenvalues of a product of two real symmetric matrices real?
Under what conditions are the eigenvalues of a product $M = A B$ of two real symmetric matrices $A$ and $B$ real?
And is there a way to relate the signs of the eigenvalues of $M$ to any properties of $...
- 61
3
votes
1
answer
309
views
Eigenvalues two-fold degenerate
Consider the matrix $$A:=\left(
\begin{array}{cccc}
0 & a & 0 & 0 \\
f & 0 & b & 0 \\
0 & e & 0 & c \\
0 & 0 & d & 0 \\
\end{array}
\right)$$
I ...
5
votes
2
answers
343
views
Maximal eigenvalue of a correlation matrix with some entries fixed as zeros
Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
- 51
20
votes
6
answers
42k
views
Eigenvalues of symmetric tridiagonal matrices
Suppose I have the symmetric tridiagonal matrix:
$$ \begin{pmatrix}
a & b_{1} & 0 & ... & 0 \\\
b_{1} & a & b_{2} & \ddots & \vdots \\\
0 & b_{2} & a & \...
4
votes
1
answer
187
views
largest eigenvalue of the difference between two quadratic forms
Let $U,V\in\mathbb{R}^{4\times n}$ such that $UU^T=VV^T=I$, and $A\in\mathbb{R}^{n\times n}$ be an Hermitian matrix.
Is it true that
$$\sqrt{\lambda_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}...
- 503
3
votes
2
answers
394
views
Eigenvalues and eigenvectors of non-symmetrical tridiagonal matrix
The question is the following: given a matrix
$$A=\begin{pmatrix}
1& 2 & & & & \\
1& 0& 1 & & & \\
& 1& 0& 1 & &\\
& &...
- 145
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
0
votes
1
answer
224
views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
-2
votes
1
answer
183
views
Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
- 25
3
votes
1
answer
5k
views
Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks
Consider the $(m+n) \times (m+n)$ block matrix
$$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$
I need references where they are talking about the relation between the eigenvalues of $M$ ...
- 1,269
1
vote
1
answer
136
views
Matrix transformation that always works?
Consider the matrix
$$A_2:= \begin{pmatrix} a & b_1 \\ b_2 & a\end{pmatrix}.$$
Let $\sigma_2 = \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}$, then
$$\sigma_2 A_2 \sigma_2 = \begin{...
3
votes
1
answer
144
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
1
vote
1
answer
209
views
Eigenvalues invariant under 90° rotation
Consider $N \times N$ matrices
$$A = \begin{bmatrix}
0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & & 0 \\
\vdots & 1 & 0 & \...
- 536
1
vote
0
answers
114
views
Higher dimensional Cauchy interlacing theorem
If $A$ is a Hermitian matrix and $A_j$ the principal minor with the $j$ row and column deleted and $\phi_A(x)$ the characteristic polynomial. The Cauchy interlacing iheorem states that the roots of $\...
- 1,362
6
votes
1
answer
601
views
Monotonicity of eigenvalues
We consider block matrices
$$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and
$$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$
Then we define the new matrix
$...
- 536
2
votes
1
answer
998
views
Diagonalizing a symmetric block matrix
Let us consider the matrix
$$ 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
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
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
1
vote
2
answers
446
views
Transforming matrix to off-diagonal form
I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$
The matrix I have is of the form
$$ C = \begin{pmatrix} 0 & a & b & ...
- 536
1
vote
0
answers
223
views
Fastest algorithm for finding the closest semi-definite matrix?
Given a real-valued, symmetric matrix $A \in \mathbb{R}^{n \times n}$, I'm interested in finding the closest positive semi-definite matrix $X^*\in \mathbb{R}^{n \times n}$:
$$
X^* = \mathop{\text{...
- 723
7
votes
0
answers
195
views
Hölder continuity of spectrum of matrices
Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
- 12.5k
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
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
-1
votes
1
answer
155
views
Companion matrices must have geometric multiplicity one, linear recurrence sequence view [closed]
I posted this question on math stackexchange weeks ago, and it have not receive an answer yet after a bounty offer...
I've been recently playing around with the linear recurrence sequences. Consider ...
- 109
1
vote
1
answer
241
views
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
3
votes
2
answers
853
views
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
6
votes
1
answer
4k
views
Minimum and maximum eigenvalue
I don't know if this is the right place to post this question, but I find it interesting and have not gotten an answer elsewhere. If it violates any rules, I will gladly delete it.
Let $\Lambda$ be ...
- 689
0
votes
0
answers
232
views
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
36
votes
2
answers
32k
views
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
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
12
votes
2
answers
2k
views
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix?
What is known about the eigenvectors of the $2^n \times 2^n$ Hadamard matrix defined recursively by $H_1=(1)$ and $$ H_N=\begin{pmatrix}H_{N/2} & H_{N/2} \\ H_{N/2} & -H_{N/2}\end{pmatrix}, $$ ...
- 1,324
2
votes
0
answers
345
views
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 $i∈{1,…,n}$ ...
- 145
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
1
answer
1k
views
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)...
