Questions tagged [positive-semidefinite]
Relating to a symmetric $n\times n $ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$
0
votes
1answer
29 views
Can i prove that this matrix is PSD?
I have matrix $A \in \mathbb{R}^{N \times N}$, such that $A(i,j)=trace(B_iCB_j), \forall ij$.
Matrices $B_i$ and C are PSD and symmetric with positive entries. Can I prove that $A$ is PSD too?
In ...
0
votes
0answers
17 views
Why can a constraint on a matrix being positive definite be rewritten as the matrix minus the identity being positive semidefinite?
My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the ...
1
vote
1answer
37 views
Why taking integral of both sides of matrix inequality is allowed?
How to show if $\nabla^2 f(x) \succeq \alpha I$, then the function is $\alpha$-strongly convex?
In my optimization notes I have
$$\nabla^2 f(x) \succeq \alpha I \rightarrow \alpha\text{-strongly ...
0
votes
1answer
31 views
How can i prove the following function is positive?
I have the following function.
$F =[x_1,x_2,...,x_n]_{1 \times n}*M_{n \times n}*([\dfrac{x_1}{|x_1|^{1/2}},\dfrac{x_2}{|x_2|^{1/2}}, ..., \dfrac{x_n}{|x_n|^{1/2}}]^T)_{n \times 1}$
Which $x \in \...
1
vote
1answer
30 views
How to show $\text{Tr}(AB) \leq \text{Tr}(AC)$ where $B \preceq C$?
Given three positive semi-definite matrices $A, B, C$. Show $\operatorname{Tr}(AB) \leq \operatorname{Tr}(AC)$ where $B \preceq C$?
This inequality is the matrix form of multiplying a positive ...
0
votes
1answer
29 views
How to prove a matrix is positive semidefinite?
Let $X\in S^3_+$ be a semidefinite cone. Show the explicit conditions on the components of $X$.
I wanted to show for a positive semidefenite matrix $X$ we have $z^T Xz\geq0\forall z$:
$$\begin{...
0
votes
1answer
21 views
Is this matrix positive semidefinite (Symmetric matrix, with particular pattern)
Let's consider a symmetric matrix A.
If for each row, the diagonal entry is equal or larger than the magnitude of any other element, that is
$$a_{ii} \geq |a_{ij}| \quad\text{for all rows } i \text{...
0
votes
1answer
44 views
Why $A\succeq B$ implies that all eigenvalues of $A$ is greater than or equal to all eigenvalues of $B$? [closed]
$A\succeq B$ means $v^TAv\ge v^TBv$ for any $v$.
4
votes
1answer
57 views
What is the formula for projection onto spectraplex?
A spectraplex (special case of spectrahedron) is the set of all positive semi-definite matrices whose trace is equal to one. Formally, let
$$
S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
2
votes
2answers
39 views
The proof of positive semi-definite for a kernel
How to prove the following kernel $K$ over $\mathbb R \times \mathbb R$ is positive semi-definite:
$$K(x_i, x_j) = e^{-\lambda[\sin(x_i - x_j)]^2},$$
where $\lambda > 0$. It looks like the gaussian ...
1
vote
1answer
21 views
If matrices $A$ and $AB$ have full column rank, how do I prove that $P_A - P_{AB}$ is positive semidefinite?
First of all, the projection matrix $P_A$ is given by $P_A = A(A'A)^{-1}A'$. Similarly, $P_{AB} = AB(B'A'AB)^{-1}B'A'$.
I have tried proving that $P_A - P_{AB}$ is itself a projection matrix, then it ...
0
votes
0answers
13 views
Positive definite squares [duplicate]
Suppose that $A, B$ are real $n\times n$ symmetric positive definite matrices such that $A - B$ is positive semi-definite. Does it follow that $A^2 - B^2$ is positive semi-definite?
2
votes
0answers
75 views
1-norm and symmetry
Define the fidelity function for positive operators by $F(\rho, \sigma) = \lVert \sqrt{\rho}\sqrt{\sigma}\rVert_1$. Here, $\lVert\cdot\rVert_1$ is the Schatten 1-norm and defined as $\lVert A\rVert_1 =...
1
vote
0answers
62 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 ...
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0answers
7 views
Nonnegative Fourier series coefficients for periodic nonnegative-definite function
Is there a simple way to show that the Fourier series coefficients of a periodic, nonnegative-definite function $\kappa$ must all be nonnegative? (By nonnegative-definite I mean that the Gram matrix $\...
0
votes
0answers
18 views
Intervals of a Multivariable Function
If the gradient at some point of a multivariable function equals $\vec{0}$, and the Hessian is positive or negative semidefinite, is there a notion, as in single variable calculus, of resolving the ...
0
votes
1answer
28 views
Positive/Negative Definite/Semidefinite Test Generality
A test to determine whether a matrix is positive definite, negative definite, positive semidefinite, negative semidefinite, or none of the above, is to calculate the determinant of every cascading ...
2
votes
0answers
59 views
Is the matrix $a_{ij} = \frac{1}{i+j-1}$ Positive Definite? [duplicate]
I have a matrix $A$ of size $n \times n$ defined as
$$a_{ij} = \frac{1}{i+j-1}$$
Where $a_{ij}$ is the $i^{th}$ row, $j^{th}$ column element of the matrix. So, $1 \leq i \leq n$ and $1 \leq j \leq n$....
1
vote
1answer
23 views
Ellipsoid representation by PSD matrix and by linear mapping
Consider the following two representations of ellipsoid:
$$E_1 = \{x \mid x^TSx\leq 1, \, S \succ 0\}$$
and $$E_2 = \{y \mid y = Ax, \, \|x\|\leq 1, \, \det(A) \neq 0\}$$
If I want $E_1=E_2$, I ...
0
votes
1answer
12 views
Prove $y^tH_f(a)y \leq 0$ with Taylors Theorem
Let the function $f \in C^2(\mathbb{R}^n;\mathbb{R})$ have a local maximum in the point $a \in \mathbb{R^n}$.
How can one prove the following with Taylor's theorem:
The following applies: $y^tH_f(a)...
0
votes
1answer
18 views
Find closet PSD matrix to given diagonal matrix
Let's say I have an arbitrary diagonal matrix $A$. I want to convert this diagonal matrix to the closest matrix by some metric that is also PSD. Is there a standard way to do this?
This is for a ...
0
votes
0answers
12 views
Sum of degenerate quadratic forms.
I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
0
votes
0answers
29 views
Inverse of Graph Laplacian does not exist in Matlab
In MATLAB, I have the following Laplacian matrix:
$$L = \begin{bmatrix} 11&-5&-6&0&0&0\\ -5&20&-8&-3&-4&0\\ -6&-8&20&0&-6&0\\ 0&-3&...
0
votes
0answers
41 views
Condition for a symmetric matrix to contain only positive entries?
I have a matrix defined as $\mathbf{X} -\mathbf{Y}\mathbf{Y}^\top$, $\mathbf{Y}$ is a matrix containing only non-negative entries and $\mathbf{X}$ is a similarity matrix, hence $\mathbf{X}$ also ...
6
votes
1answer
50 views
If a matrix $Q$ is symmetric and positie definite, is it possible to show that the matrix $Q-A^T(AQ^{-1}A^T)^{-1}A$ is also positive definite?
If I have a symmetric and positive definite $n\times n$ matrix $Q$ and a full row-rank totally unimodular $m\times n$, where $m<n$, matrix $A$, is it posible to show that the matrix
$$Q-A^T(AQ^{-...
0
votes
0answers
24 views
The Riccati equation and its asymptotic behavior
Consider matrices $A\in\mathbb{R}^{n\times n},B\in\mathbb{R}^{n\times m}$,
a positive semidefinite symmetric matrix $Q\in\mathbb{R}^{n\times n}$
and a positive definite symmetric matrix $R\in\mathbb{R}...
3
votes
2answers
47 views
Positive semidefiniteness of symmetric matrix with diagonal = 1 and non-diagonal elements less than 1
Does anyone know any useful results with respect to symmetric matrices with constant diagonals (specifically with respect to whether all eigenvalues are greater than $0$)? I am working on a set of ...
0
votes
0answers
15 views
Positive semidefinite block matrix is PSD with negated off-diagonal blocks?
I am trying to follow some notes that imply
$$
M=
\begin{bmatrix}
A&B^T\\B&C
\end{bmatrix}
\succeq 0
\Longleftrightarrow
M'=
\begin{bmatrix}
A&-B^T\\-B&C
\end{bmatrix}
\succeq 0$$
and ...
2
votes
1answer
49 views
Decomposing a positive semi-definite matrix with all -1,+1 elements
Claim.$\,$ A matrix $\,X \in \{-1,1\}^{n\times n}\,$ is positive semi-definite if and only if it is of the form $X= xx^{T}$, for some $x \in \{-1,1 \}^n$.
How can I prove this? Proving the 'if' part ...
-1
votes
1answer
45 views
Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable. [closed]
Let $A$ be a positive definite matrix and $B$ be a positive
semi-definite matrix. Then $AB$ is diagonalizable.
I want to see if this is true or false.
0
votes
0answers
20 views
Help with matrix selection in matlab
I have this equalities, with $\Delta X$ and $\Delta S$ unknown matrices, $X$, $S$ and $\tau$ known. $$W^{-1}\Delta X W^{-1}+\Delta S=\tau X^{-1}-S$$or equivalently $$\Delta X + W\Delta S W=\tau S^{-1}...
-4
votes
2answers
163 views
Prove the following set of unit vectors is orthogonal.
Suppose ${v_1 , v_2 , ……..v_n}$ are unit vectors in $\mathbb R^n$ such that $ || v||^2 = \sum_{n=1}^{\infty} | <v_i , v>|^2 $ for all $v \in\mathbb R^n $ Then I have to prove that ...
0
votes
1answer
26 views
Help with Matlab problem with positive semidefinite matrices
I need to find a positive semidefinite matrix $V$ such that
$$\delta(V)\leq\beta$$
where $$\delta(V)=||I-\frac{1}{\mu}V^2||_F\quad
\text{and }\quad\mu=\textbf{tr}(V^2).$$
Any ideas how can I get it?
...
2
votes
1answer
55 views
The true meaning of eigenvalues and eigenvectors, and positive (semi) definite matrix?
After studying some linear algebra, the true meaning of eigenvalues, eigenvectors, and positive definite matrix are still ambiguous. So, could you answer my questions or explanations about those ...
2
votes
1answer
51 views
Help with Dual problem in SDP
I'm having a problem to find the Dual of a Semidefinite programing problem:
$$\min\;\;(tr(U)+tr(V))/2$$
$$s.t.\;\; \left[ \begin{array}{cc}
U & X \\
X^T & V
\end{array} \right]\succeq0$$
$$...
0
votes
3answers
28 views
Matrix with no negative elements = Positive Semi Definite?
A matrix $A$ is positive semi-definite IFF $x^TAx\geq 0$ for all non-zero $x\in\mathbb{R}^d$. If all elements of $A$ are non-negative, does this guarantee that $A$ is positive semi-definite?
0
votes
0answers
56 views
Reformulate problem involving simultaneous diagonalization as linear program
(a) Let $\mathbb{S}^n$ be the set of $n\times n$ real symmetric matrices. Is minimizing $\lambda_{\max}(M)$ subject to $M\succeq 0$ and $\text{trace}(X_iM)=a_i$ for all $1\le i\le m$ a convex ...
0
votes
1answer
19 views
$LDL^\top$ for symmetric positive semidefinite matrices that are not positive definite
I have a symmetric positive semidefinite matrix (which is not positive definite) with integer entries and I know that I have an $LDL^\top$ decomposition for it (well mainly because Maple gives me one)....
5
votes
1answer
102 views
Trace inequality for a product of p.s.d. matrices and their pseudo inverse.
Let $A, B_i$ be positive semidefinite real matrices. Let $\dagger$ stand for the Moore-Penrose generalized inverse. I managed to prove that if
$\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
0
votes
1answer
30 views
Is a convex combination of positive semi-definite matrices itself positive semi-definite?
Suppose I have $\Sigma_1, \dots, \Sigma_n$ all positive semi-definite, and $\lambda_1, \dots, \lambda_n$ such that $\lambda_i \geqslant 0$ and $\lambda_1 + \dots + \lambda_n = 1$. Is it true that $\...
1
vote
1answer
50 views
Does $ 0 \leq A \leq B \implies 0\leq B^\dagger \leq A^\dagger$?
The question is stated in the title. Where $A$ and $B$ are positive definite real matrices and $\dagger$ stands for the Moore Penrose inverse.
This question link1 seems to be close but it has an ...
0
votes
1answer
38 views
Is $\pmb{x}<\pmb{y}$ implies $A\pmb{x}<A\pmb{y}$ when $A$ is positive definite?
I think the title is self-explanatory. I would like to have a small proof supporting your statement. $\pmb{x}$ and $\pmb{y}$ denote vectors and $A$ denote a square positive definite matrix.
0
votes
0answers
26 views
Is a generalized inverse of a s.d.p matrix also s.d.p?
Let $A \in \mathbb{R}^{n\times n}$ be a simmetric semidefinite positive matrix.
Let $A^+$ be a generalized inverse in the sense that $A A^+ A = A$. Is $A^+$ sdp?
For the Moore Penrose inverse the ...
1
vote
3answers
47 views
Is $e^{-(L+L^T)}$ a positive definite matrix?
$L$ is a nonsymmetric Laplacian matrix and the zero eigenvalue is simple, is it true that $e^{-(L+L^T)}$ a positive definite matrix (I can be sure that it is non-negative matrix)?
3
votes
1answer
138 views
Directional derivatives of matrix trace functionals
Let $P$ be an $n\times n $ positive semidefinite matrix over $\mathbb{C}$, let $p\in\mathbb{R}$ be in the range $0<p<1$. Consider the function $g:[0,\infty)\rightarrow\mathbb{R}$ defined by $g(x)...
0
votes
1answer
26 views
Hessian and gradient in a matrix
There is following matrix:
$$\begin{pmatrix}\nabla^2g(x) & \nabla g(x)\\\ \nabla g(x)^{T} & 1\end{pmatrix} \ge 0$$
the "$\ge$" is general inequality (not element-wise), meaning the matrix ...
1
vote
2answers
56 views
how to prove positive definite complex matrix
In order to prove that that a quadratic function is convex: $x^HAx$ , $A$ needs to be positive semi definite. Where $A= \Phi^H \Phi$ and $\Phi$ is a n×k matrix with possibly k linearly independent ...
1
vote
1answer
64 views
can this matrix be positive semidefinite?
Let $ S \in \mathbb{R}^n $ and $ A \in \mathbb{R}^{n \times n} $ Is it possible to find $ \alpha \in \mathbb{R} $, as a function of $A$, such that
$$ S S^T ( \alpha 1_{n \times n} + A ) + ( \alpha 1_{...
1
vote
1answer
102 views
Closed form solution for block matrix A in $AXA^T=C$
Looking for help for closed form solution for A, given C and X in the matrix equation
$AXA^T=C$
A (size $M x N$), X ($N x N$) and C ($M x M$) are (appropriately sized) block matrices
$A =
\...
1
vote
1answer
52 views
Does maximizing the Rayleigh quotient using the method of Lagrange multipliers require the matrix to be positive semidefinite?
I’m faced with the problem of maximizing a Rayleigh quotient:
$$\max_h \,\, \frac{h^t H h}{||h||^2}$$
Which is equivalent to solving
$$\max_h \,\, h^t H h $$
$$ s.t. ||h||^2=k >0, k \in \...
