User daizhuo - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T07:22:18Z http://mathoverflow.net/feeds/user/7595 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44777/how-to-characterize-real-square-matrices-a-such-that-vav-0-for-all-real-vec How to characterize real square matrices A, such that v'Av >= 0, for all real vectors v with 1'v=0 (1 is the vector of all ones)? daizhuo 2010-11-04T03:21:47Z 2013-01-01T22:19:30Z <p>I derive this question while trying to prove the monotonicity of a differentiable vector function $f(x)$ that maps from $X\subset R^n$ to $R^n$ (Here function $f(x)$ is called monotone if $(x-y)'(f(x)-f(y))\geq 0$, $\forall x,y\in X$). The domain $X$ only consists of vectors $x$ such that $1'x=0$, here $1$ is the vector of all ones.</p> <p>Using the mean-value theorem, we have that $f(x)$ is locally monotone at $x$ (namely $(y-x)'(f(y)-f(x))\geq 0$, $\forall y\in X$) if its Jacobian matrix evaluated at $x$, which we label as $A$, satisfies the following condition:</p> <p>$$v'Av\geq 0,\quad \forall v \text{ such that } 1'v=0.$$</p> <p>This is a weaker condition than positive semidefiniteness. However, while there are a number of ways to characterize positive semidefinite matrices, for example, see <a href="http://en.wikipedia.org/wiki/Positive-semidefinite_matrix#Characterizations" rel="nofollow">this Wikipedia page</a>, how can I characterize the above defined matrices?</p> http://mathoverflow.net/questions/32515/how-to-write-matlabs-dot-operators-in-mathematical-expressions How to write Matlab's dot operators in mathematical expressions? daizhuo 2010-07-19T17:20:11Z 2011-04-26T05:27:13Z <p>Matlab has a set of dot operators, such as .*, ./, .^. Each of these operators consists of a dot and a normal algebraic operator. They perform element-wise algebraic operations on a matrix. For example, consider the following codes</p> <pre><code>A = [1 2 3; 3 2 1]; x = [1 2 4]; B = A.^2 y = 1./x </code></pre> <p>The result is</p> <pre><code>B = 1 4 9 9 4 1 y = 1.0000 0.5000 0.2500 </code></pre> <p>I find these dot operators very convenient. My question is, how to write these dot operators in mathematical expressions? (By mathematical expressions, I mean the expressions used in proofs.)</p> <p>EDIT - One obvious way is to define the result matrix element-wise. But is there a way to write this result in a more compact manner?</p> http://mathoverflow.net/questions/47888/expectation-multinomial-distribution-and-monotonicity-a-conjecture Expectation, multinomial distribution, and monotonicity (A conjecture) daizhuo 2010-12-01T10:40:22Z 2010-12-01T16:26:37Z <p>Let $n$ and $k$ be two positive integers. Let $S = \{ \mathbf{p} \in \mathbb{R}^k : \mathbf{p} \geq 0, \sum_{i=1}^k p_i = 1 \}$ (i.e., a simplex). Consider a function $\mathbf{f}:\mathbb{Z}^k \rightarrow \mathbb{R}^k$. And from $\mathbf{f}$ we may define function $\mathbf{g}: S\subset \mathbb{R}^k \rightarrow \mathbb{R}^k$ as follows $$\mathbf{g}(\mathbf{p}) = \mathbb{E}[\mathbf{f}(\mathbf{X})], \quad \mathbf{X}\sim \text{Multinomial}(n,\mathbf{p}).$$ Here $\mathbf{X}$ is a random vector that follows the multinomial distribution determined by the number of trials $n$ and probabilities $p_1,p_2,\dots,p_k$.</p> <p><strong>Conjecture</strong>: If $\mathbf{f}$ is monotone, that is, $$(\mathbf{x}-\tilde{\mathbf{x}})^T (\mathbf{f}(\mathbf{x})-\mathbf{f}(\tilde{\mathbf{x}})) \geq 0\quad \forall \mathbf{x}, \tilde{\mathbf{x}} \in \mathbb{Z}^k,$$ then $\mathbf{g}$ is also monotone, i.e., $$(\mathbf{p}-\tilde{\mathbf{p}})^T (\mathbf{g}(\mathbf{p})-\mathbf{g}(\tilde{\mathbf{p}})) \geq 0\quad \forall \mathbf{p}, \tilde{\mathbf{p}} \in S.$$</p> <p><strong>Remark</strong>: The above result should hold in the following two special cases (I omit the proofs) -</p> <ol> <li><p>The function $\mathbf{f}$ is affine, namely, $\mathbf{f}(\mathbf{x})=G\mathbf{x}+\mathbf{b}$ for some matrix $G$ and vector $\mathbf{b}$.</p></li> <li><p>The function $\mathbf{f}$ is "separable", meaning that $$\mathbf{f}(\mathbf{x}) = (f_1(x_1), \dots, f_k(x_k))^T$$ for some non-decreasing scalar functions $f_1(\cdot), \dots, f_k(\cdot)$.</p></li> </ol> <p>But does it hold in the general case? </p> http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite For what k, matrix (k A - B) is positive semidefinite? daizhuo 2010-11-29T02:39:35Z 2010-11-30T07:58:22Z <p>Suppose $A$ and $B$ are two $n\times n$ real symmetric matrices. $A$ is positive semidefinite. Then for what values of real number $k$, matrix $(kA-B)$ is positive semidefinite (we write as $kA-B\succeq0$)?</p> <p>If $A$ is positive definite, we may find an $n\times n$ nonsingular matrix $D$ such that $A=D^T D$. As a result, $kA-B\succeq0$ is equivalent to $$kI\succeq (D^{-1})^TBD^{-1},$$ or $k\geq \lambda_{\max}((D^{-1})^TBD^{-1}))$.</p> <p>But how to deal with the situation when $A$ is singular (but still positive semidefinite)? I know for certain that in this case we must impose additional constraint on matrix $B$. In particular, let the columns of matrix $N$ consist of a basis of the null space of $A$, then we must have that $N^T B N \preceq 0$ (i.e., $N^T B N$ is negative semidefinite). But what is the lower bound on $k$?</p> <p>Thanks.</p> http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite/47739#47739 Comment by daizhuo daizhuo 2010-11-30T17:35:26Z 2010-11-30T17:35:26Z Thanks. In fact, $B_{13}$ must be 0. Let $P_1$ be a matrix consisting of columns that form a basis of the range of $A$. Let $P_2$ be a matrix consisting of columns that form a basis of the intersection of the kernel of $A$ and the range of $B$. Let $P_3$ be a matrix consisting of columns that form a basis of the intersection of the kernel of $A$ and the kernel of $B$. Then columns of $P=(P_1, P_2, P_3)$ form a basis of $\mathbb{R}^n$ in which $A$ and $B$ took the form you showed above. In particular, $B_{13} = P_1^T B P_3 = 0$ because $B P_3 = 0$. http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite/47642#47642 Comment by daizhuo daizhuo 2010-11-29T09:56:51Z 2010-11-29T09:56:51Z Actually $kA - B \succeq 0$ for $k \geq -1$. http://mathoverflow.net/questions/47630/for-what-k-matrix-k-a-b-is-positive-semidefinite/47642#47642 Comment by daizhuo daizhuo 2010-11-29T09:52:11Z 2010-11-29T09:52:11Z Can you be more specific on your last sentence? In particular, why do I need $\text{ker}A\subseteq\text{ker}B$? A counter example would be the following. $A = \left(\begin{array}{cc}1 &amp; 0\\0 &amp; 0\end{array}\right)$ and $B = -I$. In this case $k A - B\succeq 0$ for $k\geq 0$, but $\text{ker}A$ is larger than $\text{ker} B$. http://mathoverflow.net/questions/44777/how-to-characterize-real-square-matrices-a-such-that-vav-0-for-all-real-vec/44851#44851 Comment by daizhuo daizhuo 2010-11-05T03:34:00Z 2010-11-05T03:34:00Z Thanks for your comment. I added to my question the definition of vector function monotonicity. http://mathoverflow.net/questions/32515/how-to-write-matlabs-dot-operators-in-mathematical-expressions Comment by daizhuo daizhuo 2010-07-20T02:06:01Z 2010-07-20T02:06:01Z Thanks for the comment. I followed the link to APL; it is interesting! http://mathoverflow.net/questions/32515/how-to-write-matlabs-dot-operators-in-mathematical-expressions/32518#32518 Comment by daizhuo daizhuo 2010-07-19T17:57:25Z 2010-07-19T17:57:25Z Oh thanks for your response! But are there more compact ways to write these facts?