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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 \...
qmww987's user avatar
  • 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 ...
qmww987's user avatar
  • 91
1 vote
0 answers
64 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}=...
hos's user avatar
  • 11
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 ...
ABB's user avatar
  • 4,058
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 ...
ElectricPhysiscist's user avatar
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=\...
ABB's user avatar
  • 4,058
0 votes
1 answer
310 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}...
CereIssou's user avatar
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{...
Alec Jacobson's user avatar
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 &...
António Borges Santos's user avatar
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 ...
António Borges Santos's user avatar
1 vote
0 answers
179 views

QR algorithm for eigenvalues and eigenvectors of large symmetric matrices

I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices, My initial thought was to use Householder transformation with a Wilkinson shift ...
Daniel Belaish's user avatar
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{...
António Borges Santos's user avatar
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 ...
TobiR's user avatar
  • 103
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 $\...
CHUAKS's user avatar
  • 1,362
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& \...
Young Q's user avatar
  • 43
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)}...
Student88's user avatar
  • 503
-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}\\ \...
A. R.'s user avatar
  • 25
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 \...
shahulhameed's user avatar
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 & ...
Sascha's user avatar
  • 536
1 vote
0 answers
49 views

Bounds on Eigenvalues After Skew-Symmetric Perturbation

Consider two matrices $\mathbf{J} \in \mathbb{R}^{n \times n}$ and $\mathbf{L} = -\mathbf{L}^T \in \mathbb{R}^{n \times n}$. I am trying to upper bound the eigenvalues of their sum: $$\mathbf{A} = \...
Leo's user avatar
  • 11
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 & &\\ & &...
Connor's user avatar
  • 145
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 \...
Marin's user avatar
  • 1
-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 ...
Lab's user avatar
  • 109
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 \...
Jochen Glueck's user avatar
1 vote
0 answers
163 views

An estimation of the largest eigenvalue of a submatrix of $\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$

Let us consider the following matrix $A=(a_{k,l})$ where $$A=\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$$ Let us consider the submatrix $A_0$ of $A$ whose entries are those $a_{k,l}$ where $k\...
ABB's user avatar
  • 4,058
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 $...
Nxy's user avatar
  • 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 $M$ ...
getraparth's user avatar
2 votes
0 answers
69 views

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 \...
Leo's user avatar
  • 175
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 & \...
Sascha's user avatar
  • 536
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....
ghc1997's user avatar
  • 823
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 $$...
Pritam Bemis's user avatar
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 ...
Guido Li's user avatar
2 votes
0 answers
121 views

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] ...
Chriscrosser's user avatar
3 votes
2 answers
856 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 $m\times m$ ...
Vincent Granville's user avatar
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 ...
Guido Li's user avatar
2 votes
1 answer
999 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{...
Guido's user avatar
  • 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 ...
user479369's user avatar
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-...
Sascha's user avatar
  • 536
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 $...
Sascha's user avatar
  • 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 $...
John Doe's user avatar
0 votes
0 answers
46 views

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 ...
seg nana's user avatar
0 votes
1 answer
263 views

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 ...
seg nana's user avatar
2 votes
1 answer
74 views

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 $\...
ohana's user avatar
  • 143
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 $i∈{1,…,n}$ ...
ayr's user avatar
  • 145
1 vote
0 answers
618 views

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 ...
blldt's user avatar
  • 11
1 vote
0 answers
331 views

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 &...
E_Wijler's user avatar
2 votes
0 answers
81 views

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 ...
rostader's user avatar
  • 215
2 votes
1 answer
401 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 ...
R. Davis's user avatar
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)...
anonymousguyfromtheworld's user avatar
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 ...
WPCN's user avatar
  • 31

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