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# Questions tagged [traces]

For questions involving the trace of a square matrix, i.e. the sum of the elements on the main diagonal.

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### The trace map in étale cohomology

Trace has a pretty formal definition in a monoidal closed category in which $[V, V] \cong V^* \otimes V$ as the composition $\mathbb{Tr} : 1 \rightarrow [V, V] \cong V^* \otimes V \rightarrow 1$, ...
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### Another formula for the Schwinger term — problems with a calculation

$\DeclareMathOperator\Tr{Tr}$I have a problem with understanding the proof of Proposition 6.8 in the book ,,Elements of Noncommutative Geometry''. One can find the formulation of this proposition here ...
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### Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$

Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define $$u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}).$$ Question. What are necessary and ...
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### Different generating sets for conjugation invariants of several matrices

There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
1 vote
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### Spectrum invariant under (generalised) transpose as operator on trace class operators

For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
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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 ...
1 vote
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### Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$

I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
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### Hölder inequality inside trace

$\DeclareMathOperator\tr{tr}$Suppose we have positive semidefinite matrices $A_1, \dotsc, A_n$ and $B_1, \dotsc, B_n$ of the same dimension. Do we have a Hölder inequality for the trace of the ...
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### Tensor product of operator subalgebras and properties of the trace

Note that this question was already posted on MSE: https://math.stackexchange.com/questions/4290741/tensor-product-of-operator-subalgebras-and-properties-of-the-trace Let $V$ be a vector space and let ...
159 views

### For which representations of a Lie algebra is the induced trace form basic?

Let $\mathfrak{g}$ be a simple Lie algebra. Let $\rho$ be a representation of $\mathfrak{g}$ on a finite-dimensional vector space $E$. Consider now the bilinear form on $\mathfrak{g}$: \begin{equation}...
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### If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?

Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class. Consider its kernel $T(i,j) = \langle e_i, T e_j \rangle$ where $\{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
1 vote
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### Inequality for trace of a symmetric product?

Let $A$ be a real, positive-definite, symmetric operator on an $n$-dimensional space $V$. Write $\odot^k A$ for the action of $A$ on the symmetric power $\odot^k V$. Let $v_1,\dotsc,v_n$ be a basis ...
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### Closed paths, traces and spectra

Let $\Gamma$ be a graph. Write $A$ for its adjacency matrix. It is trivial to show that $\mathrm{Tr} A^k$ equals the number of closed walks of length k, that is, the number of $k$-step walks that ...
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### Traces and closed walks that do not close before their time

Let $A$ be the adjacency matrix of a graph. Then, as is well-known and trivial to show, $\mathrm{Tr} A^k$ equals the number of closed walks of length $k$. Is there a similar way to express (a) the ...
353 views

1 vote