# 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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### Trace map for universal bundle of Grassmannian

Let $G := G(k,V)$ denote the Grassmannian of $k$-linear subspaces in a $\mathbb{C}$-vector space $V$ of dimension $n$. Let $S$ denote the tautological bundle over $G$. There is a canonical map on ...
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### Singular value of Hadamard product

Let $A$ be an $n \times n$ random symmetric matrix with $E(A_{i j}) = 0$, $Var(A_{i j}) = 1/n$ for any $i,j$. $B$ is an $n \times n$ symmetric matrix with $B_{ii} = 0$. I need to find a upper bound of ...
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### Which operators on the trace-class operators extend to operators on Hilbert-Schmidt operators?

Let $\mathcal{H}$ be a separable Hilbert space and let $TC( \mathcal{H})$, $HS(\mathcal{H})$ be the space of trace-class operators and Hilbert-Schmidt operators on $\mathcal{H}$. Recall that these ...
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### Necessary and sufficient conditions on kernels of trace-class operators

Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$? We know that ...
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### Frobenius algebras and traces of modules

$\newcommand{\Hom}{\mathscr{Hom}}$ Let $C$ be a cocomplete closed symmetric monoidal category, and the tensor product preserves colimits in each variable; Let $A$ be a commutative algebra in $C$, ...
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### Is the secondary Euler characteristic a categorical trace?

Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...
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### Why trace is more natural than (preferred to) determinant for smooth map $f:M\to N$?

Cross-post from MSE. For a continuous map $f:(M,g)\to (N,h)$, between Riemannian manifolds $(M,g)$ and $(N,h)$ we can pullback $h$ by $f$. Most experts take the trace from this new tensor and work ...
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### Closed paths, closed trails and traces

Let $A$ be the adjacency matrix of a (non-oriented) graph $\Gamma$. Then $\textrm{Tr} A^k$ equals both the sum $\sum_i \lambda_i^k$ of $k$th powers of eigenvalues of $A$, on the one hand, and the ...
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### A bounded extension operator

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary $\partial\Omega$. Consider the harmonic extension operator $E :L^2(\partial \Omega) \rightarrow H^{1/2}(\Omega)$ which assigns ...
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1 vote
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### Computational complexity of computing the trace of a matrix product under some structure

I have two problems related to computing some trace, and some (possibly suboptimal) answers. My question is about a potential more efficient algorithm for each one. (More interested in an answer to ...
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### Why are traces an analogue to integrals?

In Poincare duality for singular cohomology, one integrates cohomology classes against a fundamental class to get a number $\int_{[M]} \omega$. In the formulation of Poincare duality in etale ...
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