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)