Questions tagged [positive-semidefinite]
Relating to a symmetric $n\times n $ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$
655
questions
1
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1answer
17 views
Would symmetry positive semi-definite matrix always decomposable?
Given symmetry positive semi-definite matrix $A \in R^{n\times n}$. And $Det(A) \geq 0$.
Would there always exist real matrix $B$, such that $A = B \cdot B^T$?
If so why? Or why not?
2
votes
2answers
34 views
The norm $\|S-Q\|_F$ where $Q$ is orthogonal is minimised by $Q=I$
Problem:
Suppose that $S$ is symmetric and semi-positive-definite. Let $\|\cdot \|_F$ be the Frobenius norm. Show that
$$\|S-I \|_F \leq \|S-Q\|_F$$
for all orthogonal matrices $Q$, where $I$ is ...
0
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0answers
18 views
Show that there exist matrices $A$ and $B$ such that this matrix series is not positive semi-definite
This is a follow-up on a question I asked before.
Suppose that $A$ is a $k \times k$ matrix with $\|A\|_2 \leq 1$ and $B$ is a $k \times k$ positive semi-definite matrix. Consider the following sum:
...
0
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1answer
43 views
Proof of positive semi-definiteness of hat matrix product
Consider the matrices $B = \begin{bmatrix}1 &0\\-1&1\\0&1\end{bmatrix}$, $Y = \begin{bmatrix}x_1&0 & 0\\0&x_2&0\\0&0&x_3 \end{bmatrix}$, $\tilde{B} = YB$ and $T =...
0
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0answers
17 views
Mean orthogonal projection on a vector of multidimensional normal distribution
Let $X\in\mathbb{R}^n$ be a normally distributed random vector with mean zero and positive definite covariance matrix $\Sigma$, i.e. $X\sim\mathcal{N}(0,\Sigma)$. Treating $X$ as a column vector, the ...
0
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0answers
19 views
Converting semidefinite program into standard form linear program
I'm new to semidefinite programming and I'm having trouble computing what I think is a fairly simple semidefinite linear program. I need to maximize
$$\mathrm{Tr}[H X]$$
subject to some linear ...
2
votes
1answer
30 views
Show that this matrix series is positive semi-definite
$A = B^\top B$ is a $k \times k$ symmetric positive semi-definite matrix and $\|A\|_2 \leq 1$. Consider the following sum for $c$:
$$S_n = \sum_{i=0}^{n-1} (n-i) A^{i} - \frac{1}{c} \sum_{i=0}^n A^{i}...
1
vote
1answer
24 views
Squared exponential kernel with Manhatten distance does not result in positive semi-definite matrix
Here it is stated that the squared exponential covariance function
$$C(d) = e^{-(\frac{d}{V})^2},$$
where $V$ is a scaling parameter and $d$ is a distance between two points, is a stationary ...
1
vote
2answers
29 views
Construct a diagonal-1 gram matrix with given eigenvalues
I got a question concerning positive semidefinite matrices. Given a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$, we know that its trace $\text{tr}(M)$ equals to the sum of its ...
1
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2answers
34 views
In general, are functions of this form convex?
I know that quadratic functions, i.e. functions of the form $f(x) = \frac{1}{2}x^TAx+b^Tx+c$ (with $A\in\mathbb{R}^{nxn}$, $b\in\mathbb{R}^n$, $c\in\mathbb{R}$), are convex over $\mathbb{R}^n$ when A ...
0
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0answers
23 views
Hessian matrix (2x2) has trace,det=0 but a negative eigenvalue--is this matrix PSD?
I'm trying to show whether the function $f(x)=sin(\frac{\pi x_2} {2})$ is convex for $x\in \mathbb{R}^2$, where $0<x_1\le2$ and $0<x_2\le1$
When I compute the Hessian, I find $$H=\nabla^2f=\...
0
votes
0answers
14 views
Given that a matrix is PSD, can we show whether this function is strictly convex?
For this problem it's given that $Q\in \mathbb{R}^{n\times n}$ is positive semi-definite, and now I'm trying to figure out whether
$$g(x) = \sqrt{x^TQx-1}$$
is convex over $\mathbb{R}^n$
I did a ...
0
votes
0answers
24 views
Is $\min\limits_{x,y}\{\sqrt{1+x^2}-\sqrt{1-y^2}\mid C\}$ is convex problem?
I need to understand if $$\min_{x,y}\left\{\sqrt{1+x^2}-\sqrt{1-y^2}\,\middle|\,\frac{\sqrt{(x-1)^{2}+y^2}}{\sqrt{x^2+(y-2)^2}}\leq\frac12,\ |y|\leq1\right\}$$is convex problem. I found that the ...
0
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1answer
18 views
Necessary and sufficient condition for the existence of a positive operator in a finite-dimensional inner product space
Exercise from PR Halmos's Finite-Dimensional Vector Spaces, Edition 2 (Ex 8 on page 142).
If $x$ and $y$ are nonzero vectors (in a finite-dimensional inner product space) then a necessary and ...
1
vote
1answer
19 views
Semi-positivity of an Hermitian matrix
I´m triying to show that an $n\times n$ matrix $M$ such that $M_{ij}=e^{\gamma_{ij}}$ where $\gamma_{ji}=\gamma_{ij}^{\ast}$ are complex numbers is a positive semi-definite matrix.
I made a proof of ...
0
votes
1answer
27 views
Eigenvectors of a sum of PSD matrices
Lets say I have two real-valued positive-semidefinite matrices $A$ and $B$ with eigenvectors given by $v_i$ and $w_i$, respectively. Does this tell me anything about the eigenvectors of
$C = A + B$
I ...
3
votes
2answers
68 views
If a matrix is positive-semidefinite, is Hermitian, and has a trace of 1, is it idempotent?
I have a matrix $A^{n\times n}$ that is positive semi-def. and Hermitian. Also, $Tr(A) = 1$.
I want to show that $A$ is idempotent. Are these properties enough? If so, how would one show this? If not,...
0
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3answers
24 views
Show that this function is convex given that Q is positive semi-definite
For this problem it's given that $Q\in \mathbb{R}^{n\times n}$ is positive semi-definite, and now I'm trying to show that
$$f(x) = \sqrt{x^TQx+1}$$
is convex over $\mathbb{R}^n_{++}$
I've tried to ...
0
votes
1answer
21 views
Let $A$ be positive semi-definite. Show that $a_{i,i}a_{j,j} \geq |a_{i,j}|^2$, $1\leq i<j \leq n$
I was trying to do it by induction.
First let $$ A = \begin{bmatrix} a & b \\ \overline{b} & c \end{bmatrix}.$$
Since $A$ is PSD, $det(A) = ac - |b|^2 \geq 0$. So $ac \geq |b|^2$.
Now ...
0
votes
1answer
38 views
Eigenvalues of matrix product $ADA^T$
I am considering such a matrix product, namely $ADA^T$, where $A$ is an $n\times m$ matrix with $n>m$, $D$ is an $m\times m$ positive diagonal matrix. I understand the matrix product is positive ...
2
votes
0answers
76 views
Would such a matrix exist?
Could anyone help me search for a positive semidefinite matrix $\left(a_{i,j}\right)_{4\times4}$of rank 3 with $a_{i,i}=3$
and its all 3 by 3 principal minor matrices having minimal eigenvalue $\...
0
votes
0answers
10 views
Positive semidefinite sequence, estimators of the acvs
definition of positive semidefinite for a sequence $s_n$:
$\forall t_1, \cdots, t_n \in \mathbb T, \forall a_1, \cdots, a_n \in \mathbb R^*\colon$
$$
\sum_1^n \sum_k^n s_{t_j-t_k} a_j a_k \geq 0
$$
...
0
votes
1answer
27 views
Difference between two positive semidefinite matrices
Let $A, B \in \mathbb{R}^{n \times n}$ be symmetric positive semidefinite matrices and:
\begin{equation}
\lambda_{max} (A) \leq \lambda_{max} (B)
\end{equation}
Can we prove that the matrix $C = A - ...
0
votes
1answer
30 views
Question about the diagonalizability
In some textbooks it is mentioned that, if a matrix is Positive Semidefinite, then all its diagonal entries are nonnegative and all the principal submatrices of PSD obtained by removing any number of ...
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0answers
20 views
How to calculate $\max\{\frac{1}{2}\sum_{i=1}^5(1-X_{i,i+1}) \mid X \text{ PSD}, X_{ii}=1 \forall i\in [5]\}$ analytically.
I'm trying to calculate $sdp(C_5) =\max\{\frac{1}{2}\sum_{i=1}^5(1-X_{i,i+1}) \mid X \text{ PSD}, X_{ii}=1 \forall i\in [5]\}$ analytically.
I've already bounded it:
As $\begin{pmatrix}1 & -\...
0
votes
2answers
47 views
Can a symmetric positive semi-definite matrix be transformed to any symmetric positive semi-definite matrix with the same rank?
Given a symmetric positive semi-definite matrix $A\in\mathbb{R}^{n\times n}$ and denote by $rank(A)$ the rank of $A$. Let $\bar A=Q^TAQ$, where $Q\in\mathbb{R}^{n\times n}$ is an arbitrary invertible ...
0
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0answers
12 views
A globally positive semi-definite second-order polynomial can be written as a completing square?
Consider the following second-order polynomial
\begin{equation}
f(y,z)=k_1+k_2\,y+k_3\,z+k_4\,y^2+k_5\,y\,z+k_6\,z^2,
\end{equation}
where $k_i\in\mathbb{R}$ $(i=1,\cdots,6)$ are constants. Suppose ...
1
vote
2answers
169 views
A and B are real, symmetric and positive semi-definite matrices of the same order; is AB diagonalizable?
I've already proved this with B positive definite using the fact that $B^{-\frac{1}{2}}$ does exist and AB is similar to $A^{\frac{1}{2}}BA^{\frac{1}{2}}$, but since B is positive semi-definite I don'...
0
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1answer
49 views
Show that, if $A$ is PSD, then there is an optimal solution $x_i, y_j$ of the below given optimization problem such that $x_i = y_i$ for all $i$
I'm trying to prove that for the optimization problem
$$\max \{\sum_{i,j=1}^n A_{ij}<x_i,y_j> \mid x_1,\dots x_n,y_1,\dots y_n\in S^{2n-1}\}$$
that if $A$ is PSD, then there exists an optimal ...
0
votes
2answers
19 views
Positive definite implies positive semi-definite
I have read in the post, but there is something missing I'd like to know. I do believe it's not a duplicate post. I have a specified question, probably a trivial one.
First, let $(s_n)_{n\geq 0}$ be ...
2
votes
4answers
95 views
Variation of Least Squares with Symmetric Positive Semi Definite (PSD) Constraint
I am trying to solve the following convex optimization problem:
\begin{align}
& \min_{W} && \sum_{i=1}^n (\mathbf{x}_{i}^TW\mathbf{x}_{i} - y_i)^2 \\\\
& s.t. && W \succcurlyeq ...
0
votes
1answer
27 views
A matrix optimization problem that resembles a standard semidefinite program
I have a constrained matrix optimization problem as follows
\begin{align}
\max\limits_{X} \;\; &\mbox{tr}\Big( \left(C - \frac{1}{2} B R^{-1} S \right) \Lambda X^T \Big) \\
\text{subject to} \;\...
0
votes
0answers
11 views
How to prove that an inner matrix of LU decomposition of PD matrix is also PD?
A is symmetric positive definite.
Let A=L*U (lU decomposition of the matrix A)
We need to prove that B is also positive definite.
Her's what I did:
Until now I have proved that the sum of all ...
1
vote
1answer
43 views
Convex optimization under a positive semidefinite constraint involving projections
I am interested in finding the smallest operator $X$ in the Frobenius norm (also called the Hilbert-Schmidt norm)
$$\begin{array}{ll}
\text{minimize:} & \lVert X \rVert_F^2\\
\text{subject to:} ...
0
votes
1answer
21 views
Proving the difference of two matrices with similar properties is psd
I have a vector $w \in \mathbb{R}^n$ and a symmetric matrix $V \in \mathbb{S}^{n}$.
I have the following properties:
$w_i \geq 0$ for all $i=1,\ldots,n$
$\sum \limits_{i=1}^n w_i = 1$
$V_{i,j} \geq ...
0
votes
0answers
16 views
Is it possible to construct a PSD matrix that has multiple null vector rows?
I know that if I generate a random matrix $\mathbf{A}\in\mathbb{C}^{n\times n}$ in MATLAB, then $\mathbf{B} = \mathbf{A}\mathbf{A}^{\dagger}\in\mathbb{C}^{n\times n}$ is a Hermitian (i.e., $\mathbf{B}^...
0
votes
1answer
16 views
matrix psd inequality, for addition
Given four matrices $A, \widetilde{A}, B, \widetilde{B} \in \mathbb{R}^{n\times d}$, if
$A^{\top} A \approx_{\epsilon} \widetilde{A}^{\top} \widetilde{A}$, $B^{\top} B \approx_{\epsilon} \...
0
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0answers
43 views
Prove that $A$ is positive semidefinite iff $\alpha \geq 1$
Given a symmetric matrix $n\times n$
$$A_{i,j} = \begin{cases} \alpha & \text{ if } i=j\\ 1 & \text{ if } i \neq j \end{cases}$$
Prove that $A$ is positive semidefinite iff $\...
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0answers
14 views
What does edge-transitive imply for the adjacency matrix of graph?
I'm trying to prove that if a matrix $G$ is edge transitive, we can say that
$$\vartheta(G) = \frac{n\lambda_{min}(A_G)}{\lambda_{max}(A_G) -\lambda_{min}(A_G)}$$
Where $\vartheta(G)$ is defined in ...
0
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0answers
11 views
compare variances of ols regression function
Let $\tilde{f}$ denote the ols estimation of $f(x) = \beta_0 + \beta_1x_i$. Let $\hat{f}$ denote the ols estimation of $f(x) = \beta_0+\beta_1x_i+\beta_2x_i^2$. I wish to compare the variances of $\...
1
vote
0answers
42 views
Why is the theta number / Lovasz number additive on disjoint graphs?
Let $G=(V,E)$ be the disjoint union of two graphs$G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$. That is, $V_1$ and $V_2$, $V=V_1 \cup V_2$ and $E=E_1\cup E_2$. We want to show that $$\theta(G) = \theta(G_1) + \...
1
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0answers
12 views
Is there an identity for the determinant of the sum of positive definite or semidefinite matrices?
I am familiar with the Minkowski inequality, but that is only for two matrices. What about three? four? Or in general, the determinant of the sum of matrices. Specifically positive definite or ...
0
votes
1answer
35 views
Is Positive Semi-Definiteness of a Matrix a loose measure of Independence?
I am trying to understand D-optimality criteria which tends to minimize the covariance matrix between two random variables. Now keeping that aside and in general, having PSD of a covariance matrix ...
1
vote
2answers
61 views
Moore-Penrose pseudoinverse: product on left with another matrix
The following problem comes from studying the conditional expectation of a multivariate normal distribution. Let $n\ge2$ be an integer, and let $\Sigma$ be a positive semidefinite, symmetric $n\times ...
0
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0answers
51 views
Difference of square roots of positive semidefinite matrices
Let $Y$ and $X$ be positive semidefinite matrices of size $d \times d$. We define $\sqrt{Y}$ and $\sqrt{X}$ as the unique symmetric positive semidefinite matrices that satisfy $(\sqrt{Y})^2 = Y$ and $(...
0
votes
0answers
52 views
Is the trace of the matrix exponential convex or non-convex?
I am trying to understand whether the following expression is a convex function:
$$
f\left (\mathbf{X} \right ) = \mathrm{tr}\left( \left ( \mathbf{\Lambda}+\alpha_0 \mathbf{\Psi}^T \left( I-e^\...
2
votes
1answer
30 views
Is tensor power an operator monotone function?
Let $A,B$ be postitive semi-definite operators on a finite dimensional Hilbert space. Is the following true?
\begin{equation}
A \geq B \ \Rightarrow A^{\otimes n} \geq B^{\otimes n} \quad n=1,2,3,...
\...
1
vote
1answer
111 views
Positive-definiteness of a matrix with entries $\frac1{(a_i+a_j)^\alpha}$
Let $0<a_1<\ldots<a_n$ be real numbers, and let $\alpha>0$ be given. Consider the matrix $A=\begin{pmatrix}\frac1{(a_i+a_j)^\alpha}\end{pmatrix}_{1\leq i,j\leq n}$. Then is $A$ positive-...
0
votes
1answer
67 views
Minimize the Trace of 2 PSD Matrices Product Subject to a Constraint on the Trace
Given $ A \in \mathcal{S}_{+}^{n \times n} $ (PSD Matrix) with $ \lambda_{max} \left( A \right) < 1 $ solve the following optimization problem:
$$ \arg \min_{X \in \mathcal{S}_{+}^{n \times n}} \...
0
votes
0answers
26 views
Completing matrix products to PSD matrices
Suppose matrices $A \in \mathbb{R}^{r \times n}$ and $\Sigma_1 \in \mathbb{R}^{n \times n}$ are given s.t $\Sigma_1$ is a rank-1 PSD matrix.
Are there non-trivial choices of a matrix $B \in \mathbb{...
