Questions tagged [positive-semidefinite]

Relating to a symmetric $n\times n$ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$

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Checking positive semidefiniteness in MATLAB

Let $\mathbf{A}$ be a $n\times n$ matrix. I want to check in MATLAB if it is PSD or not. Which tests, in MATLAB, should I do for this purpose? I know that if $\mathbf{A}$ is PSD then following holds ...
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Can this matrix be negative definite?

Let $d = 12$ and $m = 6$, and denote by $0_n$ and $I_n$ the zero matrix and the identity matrix of size $n \times n$. Let $D_+ \in \mathbb{R}^{m \times m}$ be a diagonal matrix with positive ...
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Semidefinite Matrix in LINGO

Using LINGO, I need to enter the following block matrix as one of my constraints $M= \left[ {\begin{array}{cc} 1 & x^T \\ x & X \\ \end{array} } \right]$ where x is an n by 1 ...
I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $... 0answers 14 views Equivalence of semidefinite decomposition? If we have an$n \times n$positive semidefinite matrix$A$and we have two decompositions such that$A = B B^T = C C^T$for some$n \times n$matrices$B$and$C$. Is it true that$B$and$C$are ... 0answers 109 views Convexity of a Log Likelihood Function Goal I would like to proof than the Negative Log Likelihood Function of Sample drawn from a Normal Distribution is convex. Below a Figure showing an example of such function: Motivation of this ... 0answers 42 views Is$E^T X E \preceq X$if$X \succ 0$and$\|E\|_F = 1$? Let$E, X \in M_n(\mathbb R)$.$X \succ 0$is positive definite and$\|E\|_F = 1$where$\|\cdot\|_F$denotes the Frobenius norm and no particular structure is assumed for$E$. I am trying to ... 0answers 74 views Nearest (with respect to weights) symmetric positive semidefinite matrix I want to compute the nearest symmetric positive semidefinite matrix, similar as Higham did. But here also weights (given by an inverse co-variance matrix) should be taken into account. So the ... 0answers 82 views Relationship between the maximum eigenvalues of a matrix and its row-manipulated form Let$A \in \mathbb{R}^{n \times n}$be a symmetric positive semi-definite matrix with sum of each row equals to zero. Is my intuition true that the following inequality always hold? $$\lambda_{\... 0answers 70 views Positive definiteness of a multivariate polynomial function Consider x \in \Bbb{R}^n and suppose f_m: \Bbb{R}^n \to \Bbb{R} which is defined as:$$f_m(x)=x^TAx+\alpha_{1}x_1^3+\alpha_{2}x_1^2x_2+\dots+\alpha_{r_1}x_n^3+\alpha_{r_1+1}x_1^4+\alpha_{r_1+2}... 0answers 34 views On the positve definiteness of a particular matrix and inverse I have the following question. Let$M=BAC=M^T$where$B\in\mathbb{R}^{m\times n}$,$C\in\mathbb{R}^{n\times m}$and$A\in\mathbb{R}^{n\times n}$invertible. Suppose$M$is positive definite,$B$is ... 0answers 22 views Is there a generalization of Araki-Lieb-Thirring inequality for four matrices? It is known that $$Tr[(AB)^n] \leq Tr (A^n B^n)$$ for$A$,$B$positive semi-definite matrices. I am looking of a generalization of it for a product of$4$positive semi-definite matrices of the the ... 0answers 73 views Gram matrix in a non-Euclidean space The Gram matrix of a set of vectors$v_1,\dots,v_n$with the usual Euclidean dot product is positive semidenifite. Suppose we have a symmetric matrix$A$that is not positive semidefinite. Can we ... 0answers 77 views Laplace transform of a time varying positive semidefinite matrix Consider a real, symmetric matrix$M(t)$with time-varying elements. Each of the elements are zero for$t<0$and positive for$t\geq 0$. Assume that$M(t)\succeq 0 \;\forall t\geq 0$(i.e., it is ... 0answers 35 views Is this matrix product positive semidefinite? Let$A,B\in\mathbb{C}^{p\times p}$s.t.$A$and$B$are contractions and$I_p-AA^*$and$BB^*$both are positive semidefinite. I want to show that the matrix $$B(I_p-\sqrt{I_p-AA^*}\sqrt{I_p-AA^*}^*)B^... 0answers 39 views Weighted sums of positive semidefinite matrices are uniformly positive definite Let$$A_n = \sum_{k=1}^{n} {\bf x}_k{\bf x}_k^T $$where {\bf x}_k is a p \times 1 vector. Suppose that there is a N \in \mathbb N such that the family of matrices \{ A_n \} is uniformly ... 0answers 165 views Projection of a Symmetric Matrix onto the space of Positive Semi-Definite Matrices Consider the symmetric matrix A\in\mathbb{R}^{n\times n} and the set C of all positive semi-definite matrices in \mathbb{R}^{n\times n}. Compute the projection of A onto C under the trace inner ... 0answers 19 views Show that this “1-bordered on top and left” matrix is positive semidefinite Suppose x_i\in\mathbb R^n, i=1,...,r, n>r and x_i^Tx_j=0 \forall i\ne j. I need to show that the matrix:$$ \sum_{k=1}^rx_kx_k^T-\textbf{11}^T\succeq 0. $$It's easy to see that this is ... 0answers 71 views Solving Quadratic program for finding perfect matching in polynomial time I am working on a variant of the assignment problem. The original assignment problem is as follows: We are given a bipartite graph with 2n vertices. Each edge has a non-negative integer weight w_{i,j}... 0answers 39 views Finding bounds for a subset of the positive semidefinite cone For a finite graph G, we say a matrix B\in \mathbb{R}^{n\times n} represents G if B belongs to this set:$$\mathscr{B}=\{ B=(b_{ij}) \mid B\textrm{ is positive semidefinite with trace at most ... 0answers 46 views Is sum of inverse of max a kernel? For$X$be a nonempty set, a function$f :X\times X\to\mathbb R$is called a kernel on$X$if for all$m\in \mathbb N$and all$x_1,\cdots,x_m \in X$$$K_f\equiv\begin{bmatrix} f(x_1,x_1) & \cdots ... 0answers 47 views Sum/Product of two Infinitely Divisible matrices is Infinitely Divisible? Definition - An entry wise non-negative matrix is said to be ID if A^{or} = [a_{ij}^r] is PSD (positive semi definite) for all r \geq 0 i.e r being a non-negative real no. I need to prove or ... 0answers 129 views Relating the eigenvalues of a sum of outerproducts after applying a change of basis Let A_1 \in \mathbb{R}^{k\times k} be a symmetric and positive definite matrix. Let x_1,x_2 \in \mathbb{R}^k. Suppose I have the outer product x_1 x_1^T, from this post we know that x_1 is an ... 0answers 34 views Move an orthonormal frame towards principal axis of a p.s.d. matrix Be \mathcal{W}=\{W=(w_1|\dots|w_p)\in\mathbb{R}^{n\times p}\;:\;W^TW = I\} the set of p-dimensional orthonormal frames in \mathbb{R}^n. Consider$$ L(W) = \mathrm{tr}(MP_W) = \mathrm{tr}(MWW^T) ... 0answers 85 views Simple sufficient conditions for matrix positive semidefiniteness? I need to show that a complicated$n \times n$Hermitian matrix is positive semidefinite. I'm wondering if there are simple sufficient conditions that can be used to show this. For instance, if a ... 0answers 23 views Cholsky and Group Structure of Correlation matrices If$X$and$Y$are correlation matrices which are strictly positive definite and$A$, and$B$are their respective Cholsky decompositions. Then is $$(AB)(AB)^{\star},$$ again a strictly positive-... 0answers 59 views To Show This Is Positive Semidefinite Given that$A\in M_n(\mathbb{R})$is asymptotically stable and we define$P=\int_{0}^{\infty} e^{A^Tt}Qe^{At}dt$. I need to show that$P$is positive semidefinite if$Q$is positive semidefinite.$Q$... 0answers 141 views Expected eigenvalues of Gram matrix Suppose$x \in R^p $is drawn from some distribution$p(x)$and$ A \in R^{N \times N} $is gram matrix formed by dot product of$x_i, x_j$. What are the expected eigenvalues of$A$when$N$is finite?... 0answers 99 views positive semidefinite matrix. Let$X$be a set, and$d : X × X \to R$a metric on$X$(which means that verifies :$d(x, y) \geq 0$,$d(x, y) = 0$if and only if$x = y$,$d(x, y) = d(y, x)$and$d(x, y) \leq d(x, z) + d(z, y)$... 0answers 633 views Properties of Transition Matrix Let$P$be a markov transition matrix, that is, all entries of$P$satisfies$0\le P_{ij}\le 1$, and row sums equal to$1$. Do we have the property that:$v^T(I-P)^2v\ge 0$, for any vector$v$and I ... 0answers 26 views Semidefinite Programming Representations of two Inequalities I have two inequalities including trigonometric polynomial: $$(1) \qquad \sum_{|k| \leq n} u_k e^{i2\pi kt} \leq 1,$$ and $$(2) \qquad \sum_{|k| \leq n} u_k e^{i2\pi kt} \geq 0.$$ Where the ... 0answers 54 views Second order condition for a unconstrained minium of a function long time reader first time writer Okay my question is on the following statement: Let$f :R^n\rightarrow \mathbb{R}$be a twice differentiable function ,$\bar{x} \in Dom(f) $with$\nabla{f(\bar{x}...
Consider the following $(n+1)\times(n+1)$ real matrix $$A=\begin{pmatrix}a&p^t\\p&D\end{pmatrix},$$ where $D$ is an $n\times n$ diagonal matrix with strictly positive entries, $a>0$, and ...