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1answer
37 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 ...
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1answer
143 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 ...
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1answer
241 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 $(A,B)\mapsto i(AB-BA)$. ...
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3answers
129 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}\ ...
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0answers
184 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 ...
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1answer
405 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 = ...
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1answer
142 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 ...
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1answer
118 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 ...
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3answers
199 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 ...
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1answer
368 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 ...
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1answer
157 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 ...
0
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1answer
115 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 $B = A + \alpha \textbf{z}\textbf{z}^H$, ...
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1answer
372 views

Diagonalization of Quaternion Hermitian matrices

How do I go about diagonalizing such a matrix. I ask because I need to sort out the following problem: Let $D$ be the quaternion algebra over $\mathbb{Q}$ with $i^2 = -1, j^2 = -11, ij=-ji=k$. ...
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3answers
815 views

Solving a quadratic equation for an hermitian matrix

I am looking for a procedure to find solution(s) for a square matrix equation $H^T H = S$ where $H = H^\dagger$ is a hermitian ($n\times n$) matrix and $S$ is a given symmetric complex matrix. Due ...
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0answers
144 views

Asymptotics of arithmetic Fuchsian groups and Shimura curves.

I'm interested in what is known/expected about some families of arithmetic Fuchsian groups. Here is the simplest family that I'm interested in: Let $E = Z[\omega]$, where $\omega = e^{2 \pi i / 3}$. ...