Concavity of Spectral mean - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:57:19Z http://mathoverflow.net/feeds/question/106612 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106612/concavity-of-spectral-mean Concavity of Spectral mean Betrand 2012-09-07T15:43:26Z 2012-09-07T19:49:19Z <p>The geometric mean of two positive definite matrices $A, B$ is defined by $A\sharp B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$. The following inequality holds true $$\left(\sum_{i=1}^n A_i\right)\sharp \left(\sum_{i=1}^n B_i\right)\ge\sum_{i=1}^n A_i\sharp B_i$$ for positive definite matrices $A_i, B_i$, $i=1\ldots, n$. </p> <p>The spectral mean is defined by Fiedler and Ptak as $A\natural B=(A^{-1}\sharp B)^{1/2}A(A^{-1}\sharp B)^{1/2}$. Is the spectral mean also concave? That is, whether </p> <p>$$\left(\sum_{i=1}^n A_i\right)\natural \left(\sum_{i=1}^n B_i\right)\ge\sum_{i=1}^n A_i\natural B_i$$ for positive definite matrices $A_i, B_i$, $i=1\ldots, n$? </p> <p>The inequality here is Loewner order. </p> http://mathoverflow.net/questions/106612/concavity-of-spectral-mean/106625#106625 Answer by S. Sra for Concavity of Spectral mean S. Sra 2012-09-07T19:49:19Z 2012-09-07T19:49:19Z <p>Some experiments reveal that the said inequality <strong>does not</strong> hold for the spectral mean.</p> <p>Here is a random (i.e., mindless) counterexample:</p> <p>\begin{equation*} A_1=\begin{bmatrix} 29 &amp; 15\\ 15 &amp; 26 \end{bmatrix},\quad A_2=\begin{bmatrix} 5 &amp; 0\\ 0 &amp; 5 \end{bmatrix},\quad B_1=\begin{bmatrix} 4 &amp; -16\\ -16 &amp; 113 \end{bmatrix},\quad B_2=\begin{bmatrix} 18 &amp; -12\\ -12 &amp; 16 \end{bmatrix}. \end{equation*}</p> <p>With this choice, we have</p> <p>\begin{equation*} (A_1+A_2)\natural (B_1+B_2) = \begin{bmatrix} 22.3606 &amp; 0.4475\\ 0.4475 &amp; 58.3661 \end{bmatrix},\quad (A_1\natural B_1)+(A_2\natural B_2) =\begin{bmatrix} 15.2404 &amp; -2.2975\\ -2.2975 &amp; 58.5164 \end{bmatrix}. \end{equation*}</p> <p>A quick calculation shows that the difference between the two matrices is an indefinite matrix, so the alleged inequality does not hold.</p>