User betrand - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:00:21Z http://mathoverflow.net/feeds/user/24492 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128033/schur-product-partial-order Schur product, partial order Betrand 2013-04-19T01:49:54Z 2013-04-19T02:02:24Z <p>Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner partial order.</p> <p>How to prove this? Where can I find a reference?</p> http://mathoverflow.net/questions/117332/a-monotone-relation-for-s-numbers a monotone relation for s-numbers Betrand 2012-12-27T15:32:03Z 2012-12-28T10:20:41Z <p>Assume $A, B$ are self-ajoint compact operators. Is it true that $\|A+iB\|\le \|2A+iB\|$? Do we have a stronger inequality $\prod_{k=1}^ns_k(A+iB)\le \prod_{k=1}^ns_k(2A+iB)$ or even stronger one $s_n(A+iB)\le s_n(2A+iB)$, $n=1, 2, \ldots$, where $s_n$ are s-numbers?</p> http://mathoverflow.net/questions/96627/when-is-a-schur-complement-an-m-matrix/99762#99762 Answer by Betrand for When is a Schur complement an $M$-matrix? Betrand 2012-06-16T00:40:38Z 2012-12-27T22:11:49Z <p>If I understand correctly, this is not true. </p> <p>Suppose $F/A$ is an M matrix, then $-F/A$ cannot be an M matrix. If $F$ satisfies your condition, so is $\begin{bmatrix}-A &amp; B \ B^{T},-D\end{bmatrix}$.</p> http://mathoverflow.net/questions/117263/optimization-version-of-the-sylvester-equation/117265#117265 Answer by Betrand for Optimization version of the Sylvester equation Betrand 2012-12-26T15:53:28Z 2012-12-26T15:53:28Z <p>Instead of minimizing the norm, the following note proposes a conjecture on minimizing the rank of $AX-XB-C$.</p> <p>M. Lin, H. Wimmer, The generalized Sylvester matrix equation, rank minimization and Roth's equivalence theorem , Bull. Aust. Math. Soc. 84 (2011) 441-443. <a href="http://www.math.uwaterloo.ca/~m29lin/LW2011.pdf" rel="nofollow">http://www.math.uwaterloo.ca/~m29lin/LW2011.pdf</a></p> http://mathoverflow.net/questions/115675/a-curious-inequality A curious inequality Betrand 2012-12-07T03:53:55Z 2012-12-20T21:42:39Z <p>Let $r_k>0$ for $k = 1,\ldots, n$, let $\alpha_k, \beta_k\in \mathbb{R}$ be given such that $|\alpha_k|\le \beta_k\le \frac{\pi}{2}$. Suppose further that $\left|\sum\limits_{k=1}^nr_ke^{i(\alpha_k+\epsilon_k\beta_k)}\right|\le 1$ for all choices $\epsilon_k=\pm1$. How to prove $$\sum\limits_{k=1}^nr_k\le n\sin\left(\frac{\pi}{2n}\right)?$$</p> <p>The inequality is known, but the proof is rather complicated. So I am looking for a concise proof. </p> http://mathoverflow.net/questions/116906/proof-of-tracenorm-equality/116912#116912 Answer by Betrand for Proof of Tracenorm Equality Betrand 2012-12-20T20:47:28Z 2012-12-20T21:02:31Z <p>Let $X=W_1^T\Sigma W_2$ be the singular value decomposition of $X$, where $W_1, W_2$ are unitaries and $\Sigma$ diagonal matrix. Taking $V=\sqrt{\Sigma}W_1$, $U=\sqrt{\Sigma}W_2$, the minimum is achieved. </p> <p>I am not sure whether I understand your comment correctly. The other direction is easy, $\|X\|_1=\|V^TU\|_1\le \|V\|_2\|U\|_2\le (\|V\|_2^2+\|U\|_2^2)/2$, where $\|\cdot\|_1$, $\|\cdot\|_2$ denote trace norm, Frobenius norm, respectively. The first inequality follows from log majorization for singular values, the second one is by AM-GM inequality.</p> 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/105745/generalizations-of-oppenheims-inequality/106471#106471 Answer by Betrand for Generalizations of Oppenheim's inequality Betrand 2012-09-06T01:16:39Z 2012-09-06T01:16:39Z <p>This is not a generalization to other matrix classes, but replacing determinant by permanent. Actually, it is a conjecture made by Bapat and Sunder: Under the same conditions $per(A \circ B) \leq (\prod{a_{ii}})per(B)$.</p> <p>...but the following result due to Jiao [On a conjecture of H. Minc, Linear and Multilinear Algebra 32 (1992) 103–105.] couldn't surprise me more $$per(A \circ B)+per (A) per (B) \geq (\prod{a_{ii}})per(B)+(\prod{b_{ii}})per(A).$$</p> http://mathoverflow.net/questions/100173/bounding-the-minimum-entry-of-an-inverse-matrix Bounding the minimum entry of an inverse matrix Betrand 2012-06-20T20:15:52Z 2012-06-20T23:56:42Z <p>Suppose $A$ is an $n\times n$ stochastic matrix, that is, entrywise nonnegative and row sums are all $1$. If $A$ is invertible, is it true that the minimum diagonal entry of $A^{-1}$ is no larger than $1$? </p> <p>Small matrices support this claim, but for larger ones, I don't know how to (dis)prove it. </p> <p><strong>Edited</strong> I forgot to add the condition that the diagonal entries of $A$ are all zero.</p> http://mathoverflow.net/questions/99299/how-to-calculate-the-inverse-of-the-sum-of-an-identity-and-a-kronecker-product-ef/99760#99760 Answer by Betrand for How to calculate the inverse of the sum of an identity and a Kronecker product efficiently? Betrand 2012-06-15T23:50:43Z 2012-06-15T23:50:43Z <p>I assume spectral decomposition of a symmetric matrix does cost too much. Let $PXX^TP=\Lambda_1$, $QYY^TQ=\Lambda_2$ be spectral decomposition of $XX^T, YY^T$, respectively. Here $\Lambda_1, \Lambda_1$ are diagonal and $P, Q$ are orthogonal. Then $$K^{-1}=(P\otimes Q)^T(I+\Lambda_1\otimes \Lambda_2)^{-1}(P\otimes Q).$$</p> <p>$(I+\Lambda_1\otimes \Lambda_2)^{-1}$ can be read directly.</p> http://mathoverflow.net/questions/99349/woodbury-formula/99751#99751 Answer by Betrand for Woodbury formula Betrand 2012-06-15T22:26:19Z 2012-06-15T22:26:19Z <p>This is just a comment (but I could not find the comment button). Sherman–Morrison–Woodbury formula plays an important role in this paper <a href="http://www.math.uregina.ca/~chguo/GL10.pdf" rel="nofollow">http://www.math.uregina.ca/~chguo/GL10.pdf</a></p> http://mathoverflow.net/questions/129943/literature-on-exponential-of-a-quadratic-form Comment by Betrand Betrand 2013-05-07T12:00:12Z 2013-05-07T12:00:12Z As each summand is concave, so f is concave. A relevant problem is when a product of quadratic form is convex. A reference comes to me is the paper &quot;Lin, Sinnamon, A condition for convexity of a product of positive definite quadratic forms, SIAM J. Matrix Anal. Appl. 32 (2011) 457-462.&quot; http://mathoverflow.net/questions/129890/a-spectral-radius-inequality Comment by Betrand Betrand 2013-05-06T20:54:55Z 2013-05-06T20:54:55Z such that ?? in the first paragraph? http://mathoverflow.net/questions/128368/stability-of-a-matrix Comment by Betrand Betrand 2013-04-22T16:49:00Z 2013-04-22T16:49:00Z What do you mean by &quot;stability of a matrix&quot;? All eigenvalues have nonpositive real part or in the unit disk? http://mathoverflow.net/questions/126666/noncommutative-arithmetic-mean-geometric-mean-inequality-and-symmetric-polynomial Comment by Betrand Betrand 2013-04-09T14:12:22Z 2013-04-09T14:12:22Z What do you mean by &quot;The two matrix version of the my conjecture follows immediately from a stronger conjecture of Bhatia and Kittaneh that was actually recently resolved&quot;? http://mathoverflow.net/questions/105745/generalizations-of-oppenheims-inequality/106471#106471 Comment by Betrand Betrand 2013-02-06T20:27:12Z 2013-02-06T20:27:12Z This would be a big conjecture. I don't know the answer. Would you post it as a new problem. http://mathoverflow.net/questions/117567/singular-values-of-the-sum-of-a-and-at/117907#117907 Comment by Betrand Betrand 2013-01-02T22:24:02Z 2013-01-02T22:24:02Z The claimed result by S.-G. Hwang &amp; S.-S. Pyo is false; see M. Fang, A note on the inverse eigenvalue problem for symmetric doubly stochastic matrices, Linear Algebra and its Applications, 2010. http://mathoverflow.net/questions/117332/a-monotone-relation-for-s-numbers/117349#117349 Comment by Betrand Betrand 2012-12-28T19:29:06Z 2012-12-28T19:29:06Z Thanks, but how do you define determinant? If $A, B$ are positive definite matrices, we have $|\det(2A+iB)|\ge |\det(A+iB)|$; see Lemma 5 of Kh. D. Ikramov, Determinantal inequalities for accretive-dissipative matrices, J. Math. Sci. (N. Y.), 121(2004) 2458-2464. http://mathoverflow.net/questions/117263/optimization-version-of-the-sylvester-equation/117265#117265 Comment by Betrand Betrand 2012-12-28T00:30:46Z 2012-12-28T00:30:46Z You are welcome. http://mathoverflow.net/questions/117332/a-monotone-relation-for-s-numbers Comment by Betrand Betrand 2012-12-27T22:17:38Z 2012-12-27T22:17:38Z A motivation is from the scalar case, as in this article <a href="http://www.math.pku.edu.cn/teachers/yaoy/Fall2011/Fan_Hoffman1955.pdf" rel="nofollow">math.pku.edu.cn/teachers/yaoy/Fall2011/&hellip;</a> http://mathoverflow.net/questions/117332/a-monotone-relation-for-s-numbers/117349#117349 Comment by Betrand Betrand 2012-12-27T22:14:17Z 2012-12-27T22:14:17Z What if $A, B$ are positive definite? http://mathoverflow.net/questions/117332/a-monotone-relation-for-s-numbers Comment by Betrand Betrand 2012-12-27T15:48:43Z 2012-12-27T15:48:43Z they are self-ajoint http://mathoverflow.net/questions/117325/for-the-series-1-2-1-1-8-1-4-1-32-1-16-1-128-1-64-does-the-ser/117327#117327 Comment by Betrand Betrand 2012-12-27T15:33:33Z 2012-12-27T15:33:33Z I am not able to delete the answer. Help! http://mathoverflow.net/questions/116906/proof-of-tracenorm-equality/116912#116912 Comment by Betrand Betrand 2012-12-21T01:36:38Z 2012-12-21T01:36:38Z 不 客 气。 互 相 帮 忙。^_^ http://mathoverflow.net/questions/116906/proof-of-tracenorm-equality Comment by Betrand Betrand 2012-12-20T20:36:03Z 2012-12-20T20:36:03Z I think you should have put &quot;the RHS is larger than or equal to the LHS&quot;. http://mathoverflow.net/questions/116856/is-there-any-relationship-between-the-eigenvector-of-sumaabb-and-sumaabb Comment by Betrand Betrand 2012-12-20T20:27:26Z 2012-12-20T20:27:26Z I guess $A′$ means the transpose. Steven shows the eigenvectors are generally different. However, there is an interesting relation between the eigenvalues, under a mild assumption. See Corollary 2.2. of Lin &amp; Wolkowicz, An eigenvalue majorization inequality for positive semidefinite block matrices, Linear Multilinear Algebra, 60 (2012), 1365-1368.