# Questions tagged [trace]

For questions about trace, which can concern matrices, operators or functions. If your question concerns the trace map that maps a Sobolev function to its boundary values, please use [trace-map] instead.

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### Using $(e^{ik\cdot x})$ as an orthonormal basis for $L^2(\mathbb{R}^d;\mathbb{C})$ to define trace.

I know that $e^{ik\cdot x}$ are not elements of $L^2$. But I believe this is often used in quantum mechanics, and wondered if there is some justification for it. For example, I have seen the trace of ...
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### Trace of off-diagonal blocks of a positive semidefinite matrix

Consider the matrix $$A=\begin{pmatrix} A_1 & A_2 \\ A_3 & A_4 \end{pmatrix}$$ Let's suppose that $A$ is a real $n\times n$ positive semidefinite and satisfies $\|A\|\leq 1$, i.e., the largest ...
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### Understanding proof of convexity of trace function

I am trying to understand Theorem 2 ($F(A,K) := \text{tr}(A^{-r}K^{\ast} A^{-p} K)$ is convex for $p,r \geq 0$ and $p+r \leq 1$) of the paper Convex trace functions and the Wigner-Yanase-Dyson ...
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### Differential of the Trace for functions applied to unbounded self-adjoint operators

Let $f\colon\mathbb{R}\to\mathbb{R}$ be in the set of Schwartz functions (or any functions with nice enough integrating properties on the real axis), $A,M$ two (possibly unbounded) self-adjoint ...
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### PCA proof, trace move

In the PCA proof, I see this move: What is the "trace" algebra/rule that allow this trace-move?
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1 vote
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### Commutative property for Hilbert-Schmidt norm?

Let $A,B$ be linear, compact, self-adjoint and even trace-class operators. Can I bound $\|ABA^{-1}\|_{HS}$ by the norm of $\|B\|_{HS}$ somehow? (Where the bound does not depend on $\|A^{-1}\|$.) Here ...
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### $\sigma$-Weakly Continuous Bounded Linear Functional on a von Neumann Algebra is Normal

I have been working on Exercise 4 in Chapter 4 of Murphy's "$C^*$-Algebras and Operator Theory", which is as follows: Let $A$ be a von Neumann algebra on $H$, and suppose that $\tau$ is a ...
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### If $X$ is a diagonal matrix of $(N \times N)$ size, then what will be the $tr ( \frac{1}{X+\frac{1}{z}I})$?

I am trying to invert $\eta$-transform (or N-transform) numerically, for which I have $\eta(z) = \frac{1}{Nz} tr( \frac{1}{X + \frac{1}{z}I})$. But I am not sure how I can find $\eta^{-1}(z)$, as I am ...
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