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Eigenvectors of tridiagonal hermitian matrix

In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as: \begin{align*} Q_n = \begin{pmatrix} 0 & q_{1,2} & 0 & 0 & ...
Denis's user avatar
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2 votes
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147 views

An "almost" true inequality for Hermitian matrices

Let $A$ be an $N\times N$ Hermitian matrix. For $p+q$ even, consider the following inequality: $$\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} (A^q)_{ii} \geq \Big(\frac{1}{N}\sum_{i=1}^N (A^p)_{ii} \Big) \Big(\...
WunderNatur's user avatar
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0 answers
33 views

Decomposition in two hermitian squares over ring of integer of CM fields

Let $k$ a CM-field of conjugation $\bar \cdot$ and maximal totally real subfield $k^+$ and $k^{++}$ its positive part $k^{++}$. Given a totally positive element $x$ in the ring of integers of $k^+$, ...
user70925's user avatar
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5 votes
2 answers
366 views

Smallest eigenvalue of a certain Toeplitz Hermitian matrix

Let $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1+i(k-\ell)\varepsilon}$ (here $i=\sqrt{-1}$ while $k,\ell$ are indices). For example, if $n=3$ we obtain $$ G=\begin{bmatrix}1 & \...
PIII's user avatar
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-2 votes
1 answer
310 views

How to compute the spectral norm of this matrix [closed]

Consider $$\left\|2\sum_{i<j}L_{ij}+4\sum_i \operatorname{diag}e_i \right\|,$$ where (1) $L_{ij}=\operatorname{diag}e_i+\operatorname{diag}e_j-e_ie_j^T-e_je_i^T$ (2) $e_i$ denotes $n$-by-$1$ vector ...
tony's user avatar
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5 votes
1 answer
291 views

Questions about hermitian positive semidefinite matrices

Motivation: I am faced with a $5 \times 5$ hermitian positive semidefinite matrix, depending on parameters, and I wish to show that it is positive definite, for any points in the parameter space (I ...
Malkoun's user avatar
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1 vote
0 answers
31 views

How does configuration or phase space change in pseudo-Hermitian (or just non-Hermtiian) QM vs Hermitian QM?

I was wondering if there is some relaxation of the configuration (or phase) space when considering pseudo-Hermitian physical situations vs Hermitian? For instance in "$C^*$-Algebras of Energy ...
kreitz's user avatar
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2 votes
0 answers
105 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
0 answers
260 views

Inequalities involving traces of products of hermitian positive semidefinite matrices

$\DeclareMathOperator{\tr}{tr}$ Fix an integer $n \geq 2$. Let $A_1, \dotsc, A_n$ be hermitian positive semidefinite matrices, with each $A_i$ being $m$ by $m$. Consider the symmetric group $S_n$ on $...
Malkoun's user avatar
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0 answers
61 views

An inequality regarding operator concave function

Crossposted from math.SE Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $...
RKC's user avatar
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1 vote
1 answer
347 views

Spectral theorem and diagonal expansion for self adjoint operators

Asked by a physicist: In physics we use the fact that, given a self adjoint operator $A$, every element in Hilbert space can be written as "generalized linear combination" of the eigenstates ...
Rosario's user avatar
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4 votes
2 answers
658 views

Do any two hermitian matrices A and B commute with the support of their commutator?

Let $A$ and $B$ be Hermitian matrices. Let $[A,B] = AB - BA$ be their commutator and let $[A,B]^+$ be the Moore-Penrose pseudoinverse of $[A,B]$. Is it then true that $A$ and $B$ both commute with the ...
saolof's user avatar
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1 vote
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Is there any property for the eigenvalues of an Hermitian matrix on which a well-structured binary mask has been applied?

While working on a quantum-focused article, I came accross the following problem. Let $\rho$ be a positive, semi-definite, $2^{n+m}$-Hermitian matrix with unit trace ($\rho$ is a density matrix). Let $...
Tristan Nemoz's user avatar
3 votes
1 answer
409 views

Derivative of eigenvalues of a symmetric tridiagonal matrix built via the Lanczos-Arnoldi scheme

Suppose $\mathbf{A}(\mu)$ being a symmetric positive definite matrix of dimension $n$ where its elements depend parametrically on the real parameter $\mu$. Suppose now to build the orthonormal basis ...
wolfram's user avatar
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0 answers
68 views

Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices

While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices. To ...
Malkoun's user avatar
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5 votes
0 answers
125 views

Complete Hermitian manifolds with vanishing Chern curvature

An old theorem going back to Boothby states that a compact Hermitian manifold with Chern curvature vanishing identically is a compact quotient of a complex Lie group with a left invariant metric. Are ...
GradStudent's user avatar
1 vote
0 answers
186 views

finding automorphisms of binary hermitian forms

Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Write $\mathcal{H}(\mathcal{O}_K)$ for the set of hermitian 2 x 2 matrices with entries in $\mathcal{O}_K$. Now, we ...
ersin's user avatar
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2 votes
1 answer
1k views

Derivative of eigenvectors of an Hermitian matrix

In the question "Derivative of eigenvectors of a matrix with respect to its components", Liviu Nicolaescu has provided an answer valid for a real matrix. As outlined in the following, the ...
S.B.'s user avatar
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1 vote
1 answer
132 views

Embedding Hermitian forms into Hilbert spaces

Let $H$ be a nondegenerate, not positive definite, Hermitian form on a complex vector space $V$ such that $$|H(x,y)|\le u(x)u(y)\tag{B}$$ for some map $u:V\to R_+$ with $u(\lambda x)=|\lambda|u(x)$ ...
Arnold Neumaier's user avatar
3 votes
0 answers
216 views

Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators

Let $A$ and $B$ be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for $e^A e^B$ as $e^C=e^A e^B$, for $C$ in the Lie algebra generated by $...
user149918's user avatar
10 votes
1 answer
256 views

Do matrices of the form $A^\ast A$ where $A$ has entries in $R\subset\Bbb C$ account for all positive semidefinite matrices with entries in $R$?

Let $R$ be a subring of $\Bbb C$ closed under complex conjugation and let $P$ be an $n\times n$ positive semidefinite matrix with entries in $R$. I'm curious if it is always possible to factor $P$ as $...
Brian Fitzpatrick's user avatar
5 votes
1 answer
314 views

Is the motivic homotopy spectrum of Hermitian K-theory $\eta$-complete?

Theorem 1.2 of the paper "The motivic Hopf map solves the homotopy limit problem for K-theory" (see https://elibm.org/article/10011880) says that (under certain assumptions on the base field) the ...
Mikhail Bondarko's user avatar
0 votes
0 answers
101 views

References for a proof or interpretation of deficiency indices theorem (von Neumann)

I am looking for a proof or some interpretation around why the domain of the new extension $D(A_U)$ in the Theorem below is given by its specific formula. I have already searched in papers and here ...
curiosity96's user avatar
4 votes
0 answers
135 views

Homogenous Hermitian form on the KLR algebra

Does there exist a homogenous conjugate-linear automorphism of the KLR algebra? I want to be able to define a homogenous Hermitian form on the (Specht) modules of the (cyclotomic) KLR algebra.
Chris Bowman's user avatar
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2 votes
0 answers
195 views

Equality condition for Araki–Lieb–Thirring inequality

I'd like to have the equality condition in the Araki–Lieb–Thirring inequality $$\operatorname{Tr} [(BAB)^r]\leq \operatorname{Tr} [(B^{r}A^{r}B^{r})],$$ valid for $A,B$ semidefinite positive and $r\...
MarcO's user avatar
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2 votes
0 answers
218 views

Integral with product of two infinite sums

I am looking for references and results on integrals with product of two infinite sums: $$S_1=\int_0^{\infty} \sum_{n=0}^{\infty} f(nx) \sum_{k=0}^{\infty} \overline{ g(kx)} dx$$ Above integral is ...
Bertrand's user avatar
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1 vote
1 answer
220 views

positive real matrix-valued function as linear combination of positive-real functions

In my question I am considering $z$ as the complex variable in the $z$-transform $X(z)$ of a discrete-time sequence $x[n]$: I have $M$ square complex matrices $\mathbf{R}_m$ of size $N\times N$. I ...
Ernest's user avatar
  • 11
1 vote
1 answer
493 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)-(...
Alpha001's user avatar
  • 143
3 votes
1 answer
182 views

Hermitian forms over $K\times K$

Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$. When $V$ is a free module, ...
Anupam Singh's user avatar
1 vote
1 answer
125 views

Diagonalisation of invariant hermitian forms and irreducible representations of tori actions

here is my question: Suppose that the torus $T^n = (S^1)^n$ acts on $\mathbb{C}^n$ by linear transformations $$(e^{i \theta_1},...,e^{i \theta_n}).(z_1,...,z_n) = (e^{i \theta_1}.z_1,...,e^{i \...
BrianT's user avatar
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2 votes
0 answers
315 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 ...
groupoid's user avatar
  • 620
1 vote
0 answers
81 views

Isotropy of skew-Hermitian forms over division algebras

Assume char(F) $\neq$ 2. Let $D$ be a central division algebra over a field $F$ and $h: V \rightarrow D$ be an anisotropic skew-Hermitian form. We can easily see that $h_{\bar{F}}$ is totally ...
Mr.Mysterious's user avatar
2 votes
1 answer
335 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 ...
Tina's user avatar
  • 21
0 votes
1 answer
131 views

On the spectrum of Hermitian matrices [closed]

I'm working on the adjacency matrix of some graphs and need some facts about Hermitian matrices which have exactly two distinct eigenvalues. Can anybody help me introduce source about spectrum of ...
A. Mpi's user avatar
  • 351
3 votes
0 answers
181 views

Diagonalization of Hermitian Matrix Polynomials

I have a question on the decomposition of polynomial matrices. Suppose $A(\lambda) = \sum_{j=0}^L \lambda^j A_j$ is an $n \times n$ matrix of polynomials, which is Hermitian on the real axis $\lambda ...
William's user avatar
  • 31
2 votes
0 answers
76 views

Conditions on a $n\times n$ Hermitian matrix such that its extremal eigenvectors have equal magnitude entries

Is it possible to find (necessary and sufficient) conditions on a general $n\times n$ Hermitian matrix $A$, such that its extremal eigenvectors (the eigenvectors corresponding to the maximum and ...
jvn99's user avatar
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3 votes
0 answers
104 views

Restriction of a singular metric with minimal singularities

Let $X $ be a smooth complex algebraic variety and $L $ a pseudo-effective line bundle on $X $, consider $h $ to be a singular Hermitian metric with minimal singularities on $L$ and $|A|$ be the ...
Joaquín Moraga's user avatar
1 vote
1 answer
210 views

Hermitic connections on complex line bundles with imaginary curvature form

It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
Alex M.'s user avatar
  • 5,357
1 vote
1 answer
229 views

Hermitian Projections on $C[0,1]$

If $X$ is a normed linear space and $S(X)$ its unit sphere, $X′$ its dual space and $Π=\{(x,f)∈S(X)×S(X′) \ | \ f(x)=1\}$, then for an operator $T$ on $X$, the numerical range $V(T)$ is defined by $V(...
mselcuk's user avatar
  • 35
0 votes
1 answer
325 views

Meaning of $[A,B]$ when $A$, $B$ are self-adjoint

This is just a question about notation, but it got no useful answers on math.stackexchange. Let $L$ be the Lie algebra of $n\times n$ Hermitian matrices, with Lie bracket $(A,B)\mapsto i(AB-BA)$. ...
Steven Landsburg's user avatar
1 vote
3 answers
206 views

Extending GUE to a measure on operators?

Let $\mathcal H_n$ denote the space of Hermitian $n\times n$ matrices and let $\mu_{GUE}$ denote the following measure on $\mathcal H_n$: $$ \mu_{GUE}(dM) = \exp\left( -\frac{n}{2}\text{tr}\ M^2\right)...
pre-kidney's user avatar
  • 1,299
1 vote
0 answers
447 views

Bound the expectation of trace norm of random Hermitian matrix

Suppose $H_i$ are traceless $d\times d$ Hermitians, $X_i$ are Standard normal distribution for $1\leq i\leq d^2$. We would like to bound the following expectation on the trace norm $\mathbb{E}|\sum_{...
gondolf's user avatar
  • 1,503
4 votes
1 answer
580 views

Affine space structure on the space of Hermitian connections

I'm reading Gauduchon's paper Hermitian connections and Dirac operators. For a fixed almost-Hermitian manifold $(M, g, J)$ let $\mathcal A(g, J)$ be the space of connections $\nabla$ s.t. $\nabla g = ...
David P's user avatar
  • 585
1 vote
1 answer
179 views

Which $\frak{sl}_2$-Representations Arise From Hermitian Metrics

Recal that $\frak{sl}_2$ is the Lie algebra with basis elements $e,f,h$, and bracket $$ [e,f] = h, ~~~ [h,e] = 2e, ~~~ [h,f] = -2f. $$ For $M$ a $2n$-complex manifold, the Lefschetz identities tell us ...
Christian Fischmann's user avatar
1 vote
1 answer
5k views

Largest eigenvalue of the sum of Hermitian matrices [closed]

Is there an expression for the largest eigenvalue of the sum of two Hermitian matrices in terms of the spectrum of the same matrices?
Benjamin's user avatar
  • 2,069
2 votes
1 answer
134 views

Projecting solutions of Hermitian forms over local rings

Let $R$ be a local ring (commutative and with $1$) with maximal ideal $M$, with an involution $\theta$. Let $h$ be a Hermitian form on $R^n$, i.e. $h:R^n\times R^n\rightarrow R$ such that $h$ is $R$-...
Erik Rijcken's user avatar
3 votes
3 answers
365 views

Is it possible to find $h$ hermitian metric such that $Isom_{h}(X) \cong Aut(X)$?

I suspect this is true for some class of analytic manifolds (Riemann surfaces maybe), but my knowledge in differential geometry is very poor, so I could not conclude it. For complex manifolds, is it ...
user40276's user avatar
  • 2,209
6 votes
1 answer
650 views

Diagonalization for sums of Hermitian matrices

I found an interesting question about diagonalizable matrices, Let $A,B\in \mathcal{M}_n(\mathbb{C})$ Hermitian, such that $AB\neq BA$. Do there exist complex numbers $u\neq v$, such that $A+uB$ and ...
user avatar
-3 votes
1 answer
269 views

Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)

For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ? To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
dehiker's user avatar
  • 101
0 votes
1 answer
329 views

Norm bound on eigen-vector change caused by rank-one update

Suppose $A$ is a positive semi-definite, Hermitian matrix with a unit-norm eigen vector $\textbf{v}$ corresponding to its largest eigen value $\lambda$. Let $B = A + \alpha \textbf{z}\textbf{z}^H$, ...
Arun's user avatar
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