Kania, Tomasz
(2015)
*A short proof of the fact that the matrix trace is the expectation of the numerical values.*
American Mathematical Monthly, 122 (8).
pp. 782-783.
ISSN 0002-9890

PDF (1402.4272v1)

1402.4272v1.pdf - Accepted Version

Available under License Creative Commons Attribution.

Download (75kB)

1402.4272v1.pdf - Accepted Version

Available under License Creative Commons Attribution.

Download (75kB)

Official URL: https://doi.org/10.4169/amer.math.monthly.122.8.78...

## 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 $\mathbb{C}^n$.

Item Type:

Journal Article

Journal or Publication Title:

American Mathematical Monthly

Uncontrolled Keywords:

/dk/atira/pure/subjectarea/asjc/2600

Subjects:

Departments:

ID Code:

73792

Deposited By:

Deposited On:

18 Jun 2015 05:45

Refereed?:

Yes

Published?:

Published

Last Modified:

09 Aug 2022 00:01