Talk:Trace inequality

General operator inequalities

Some inequalities, such as Löwner–Heinz theorem and Effros's theorem does not involve directly the trace function. It would not be better to have a general 'Operator inequalities' page and move these sections to there? — Preceding unsigned comment added by Saung Tadashi (talk • contribs) 01:12, 26 May 2015 (UTC)

Proof of Klein's inequality

In the proof of Klein's inequality, it is said that ${\tfrac {\phi (t)-\phi (0)}{t}}$ is monotone decreasing with respect to t - but by convexity of φ, it should be monotone non-decreasing. 134.87.176.11 (talk) 13:53, 26 December 2015 (UTC)

Mistake in Von Neumann's trace inequality

At the end of the section "Von Neumann's trace inequality" one finds the statement "The equality is achieved when $A$ and $B$ are simultaneously unitarily diagonalizable (see trace)." This is not correct; for this choose $2\times 2$ matrices

$A={\begin{pmatrix}1&0\\0&-1\end{pmatrix}},B={\begin{pmatrix}2&0\\0&1\end{pmatrix}}\,.$

These matrices are diagonal in the same basis but the stated trace inequality is not saturated:

$\left|\operatorname {Tr} (AB)\right|=1<3=\sum _{i=1}^{2}\alpha _{i}\beta _{i}$

Of course the original statement

$\sup _{U,V{\text{ unitary}}}|\operatorname {tr} (UAVB)|=\sum _{i=1}^{n}\alpha _{i}\beta _{i}$

from the cited source still holds and should maybe be stated explicitely in order to fix this? Frederik vE (talk) 14:26, 3 April 2019 (UTC)

It doesn't seem that the counter-example is valid, since in this case $\alpha _{1}=1,\alpha _{2}=-1,\beta _{1}=2,\beta _{2}=1$, and thus

$\sum _{i=1}^{2}\alpha _{i}\beta _{i}=2-1=1$

A proof that simultaneously unitarily diagonalizable is necessary and sufficient for attaining equality is given in: Miranda, Héctor. "Optimality of the trace of a product of matrices." Proyecciones. Revista de Matemática 18.1 (1999): 71-76. - Saung Tadashi (talk) 20:17, 3 April 2019 (UTC)

You are confusing eigenvalues and (the necessarily non-negative) singular values of a matrix; the section currently states von Neumann's original result regarding the singular values $\alpha _{j},\beta _{j}$ of two arbitrary complex matrices as an upper bound for the (absolute value of their) trace---here equality in general does not hold if $A$ and $B$ are diagonal in the same basis (unless these matrices also are positive semi-definite and their eigenvalues are sorted accordingly).

On the other hand von Neumann's result readily implies that for hermitian matrices $A$, $B$ with eigenvalues (not singular values so these may be negative!) $\alpha _{1}\geq \alpha _{2}\geq \ldots$, $\beta _{1}\geq \beta _{2}\geq \ldots$ one has $\operatorname {tr} (AB)\leq \sum \nolimits _{j=1}^{n}\alpha _{j}\beta _{j}$ with equality in the case of simultaneous unitary diagonalizability.

Sadly, it seems that the manuscript you linked is not available on the publication page as the PDF seems to be broken in some way. Frederik vE (talk) 11:42, 4 April 2019 (UTC)

Oops, you are right. I made a mistake and used the eigenvalues instead of the singular values. I found a direct link for the PDF of this manuscript: http://www.revistaproyecciones.cl/article/download/2748/2317.

Here the result is stated for real matrices (but maybe it can be generalized to complex matrices with minor modifications) and the hypothesis is that $A$ and $B$ share the same singular value decomposition

instead of simultaneous unitary diagonalizability. - Saung Tadashi (talk) 13:03, 4 April 2019 (UTC)

Sadly, I can neither retrieve the PDF from the original page (problem with my browser?) nor from the link you sent me--so I would be very happy if you could send me the PDF via e-mail

Either way I proposed an edit to the section in question as I found the result on hermitian matrices and their eigenvalues I mentioned (in the book of Marshall Olkin, which really only links the original results--those date back to the 50s and 60s). Frederik vE (talk) 14:26, 4 April 2019 (UTC)

Sorry for my delay, I sent the paper today to your email. Thanks for your edit and your correction 🙂 Saung Tadashi (talk) 02:12, 7 April 2019 (UTC)