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A short proof of the fact that the matrix trace is the expectation of the numerical values

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A short proof of the fact that the matrix trace is the expectation of the numerical values. / Kania, Tomasz.

In: American Mathematical Monthly, Vol. 122, No. 8, 10.2015, p. 782-783.

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Kania, Tomasz. / A short proof of the fact that the matrix trace is the expectation of the numerical values. In: American Mathematical Monthly. 2015 ; Vol. 122, No. 8. pp. 782-783.

Bibtex

@article{46c729bba59e40209158fc81c1669a1d,
title = "A short proof of the fact that the matrix trace is the expectation of the numerical values",
abstract = "Using the fact that the normalised matrix trace is the unique linear functional $f$ on the algebra of $n\times n$ matrices which satisfies $f(I)=1$ and $f(AB)=f(BA)$ for all $n\times n$ matrices $A$ and $B$, we derive a well-known formula expressing the normalised trace of a complex matrix $A$ as the expectation of the numerical values of $A$; that is the function $\langle Ax,x\rangle$, where $x$ ranges the unit sphere of $\mathbb{C}^n$.",
keywords = "matrix trace, numerical values, unitary matrix",
author = "Tomasz Kania",
year = "2015",
month = oct,
doi = "10.4169/amer.math.monthly.122.8.782",
language = "English",
volume = "122",
pages = "782--783",
journal = "American Mathematical Monthly",
issn = "0002-9890",
publisher = "Mathematical Association of America",
number = "8",

}

RIS

TY - JOUR

T1 - A short proof of the fact that the matrix trace is the expectation of the numerical values

AU - Kania, Tomasz

PY - 2015/10

Y1 - 2015/10

N2 - Using the fact that the normalised matrix trace is the unique linear functional $f$ on the algebra of $n\times n$ matrices which satisfies $f(I)=1$ and $f(AB)=f(BA)$ for all $n\times n$ matrices $A$ and $B$, we derive a well-known formula expressing the normalised trace of a complex matrix $A$ as the expectation of the numerical values of $A$; that is the function $\langle Ax,x\rangle$, where $x$ ranges the unit sphere of $\mathbb{C}^n$.

AB - Using the fact that the normalised matrix trace is the unique linear functional $f$ on the algebra of $n\times n$ matrices which satisfies $f(I)=1$ and $f(AB)=f(BA)$ for all $n\times n$ matrices $A$ and $B$, we derive a well-known formula expressing the normalised trace of a complex matrix $A$ as the expectation of the numerical values of $A$; that is the function $\langle Ax,x\rangle$, where $x$ ranges the unit sphere of $\mathbb{C}^n$.

KW - matrix trace

KW - numerical values

KW - unitary matrix

U2 - 10.4169/amer.math.monthly.122.8.782

DO - 10.4169/amer.math.monthly.122.8.782

M3 - Journal article

VL - 122

SP - 782

EP - 783

JO - American Mathematical Monthly

JF - American Mathematical Monthly

SN - 0002-9890

IS - 8

ER -