Questions tagged [positive-semidefinite]
Relating to a symmetric $n\times n $ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$
2
votes
1answer
51 views
Easiest way to show positive semi-definite equivalence
For an $x \in \mathbb{R}^n$, and $n$-by-$n$ identity matrix $I_n$, we are given that
$$ \begin{pmatrix} I_n & x \\ x^T & 1 \end{pmatrix} \succeq 0.$$
What is the easiest way to show that
$$ \...
0
votes
0answers
8 views
Absolute convex hull of rank 1-correlation matrices?
Does there exist a ''universal'' constant, $c > 0$ say, such that for any(!) $k \in \mathbb{N}$ every(!) $k \times k$-correlation matrix $\Sigma$ can be written as $\Sigma = c\Theta$, where $\Theta$...
1
vote
2answers
105 views
Is a square zero matrix positive semidefinite?
Does the fact that a square zero matrix contains non-negative eigenvalues (zeros) make it proper to say it is positive semidefinite?
1
vote
2answers
42 views
Proving the difference of two matrices is PSD
Claim: For $x\in \mathbb{R}^n$, we have $\operatorname{Diag}(x) - xx^T \succeq 0$ if and only if $x_i \geq 0 \ \forall i\in [n]$ and $\sum_{i} x_i \leq 1$.
Where $\operatorname{Diag}$ denotes the ...
1
vote
1answer
12 views
Negating off-diagonal blocks retains positive-semidefiniteness?
I am trying to follow some notes that state
$$
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$$
and ...
0
votes
1answer
18 views
Nearest positive semi definite matrix to a complex valued Hermitian matrix
How would you find the nearest (via Hilbert distance), PSD matrix (with trace = 1) to a Hermitian matrix? I found an answer to a similar question here. However, as I understand, Hingham's work only ...
1
vote
2answers
41 views
If both $A-B$ and $B-A$ are positive semidefinite, then $A = B$
Let $A, B$ be two positive semidefinite matrices. Prove that if both $A-B$ and $B-A$ are positive semidefinite, then $A = B$.
I can show that their diagonal elements are the same but for others I ...
0
votes
1answer
59 views
How to prove the logistic loss function is strongly convex?
The logistic loss function is:
$$\mathcal{L}=\frac{1}{n}\sum_{i=1}^n\log(1+\exp(-y_ix_i^T\theta))$$
in which $y_i\in\{-1,+1\},x\in \mathbb{R}^d$. How to show that $\mathcal{L}$ is strongly convex.
My ...
3
votes
1answer
57 views
Matrices Inequality Proof
Recently, I read a paper and there is a step which turns out not obvious to me. The statement is as follows:
All matrices here are real matrices. $F$ is an arbitrary square matrix. $\Psi$ is a ...
0
votes
0answers
17 views
Non-Negative Vs Positive Semi Definite
A matrix is PSD if $$\langle Ax, x\rangle \ge 0, \forall x \in H$$
Where, H is a hilbert space and A is a mapping $H \rightarrow H$. Is it the same as being Non-negative? I couldn't seem to find a ...
2
votes
2answers
53 views
Parametrization of unitary matrices
Does anyone know a simple way to parametrize the space of $n\times n$ complex unitary matrices into a set of independent complex numbers in some complex-rectangle? Specifically the mapping and inverse ...
0
votes
0answers
10 views
Semidefinite Matrix in LINGO
Using LINGO, I need to enter the following block matrix as one of my constraints
$
M=
\left[ {\begin{array}{cc}
1 & x^T \\
x & X \\
\end{array} } \right]
$
where x is an n by 1 ...
0
votes
2answers
30 views
Sum of symmetric, positive semidefinite matrices
Let $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{p \times n}$. Show that $A^{T}A+ B^{T}B$ is invertible if and only if $\ker A \cap \ker B =\lbrace 0 \rbrace$.
I could show that if it's ...
0
votes
0answers
27 views
Why would sequential quadratic programming fail to find global minimum?
I have a data set. A matrix $X$, $1300 \times 20$ and output vector $\mathbf{y} \in \Bbb R^{20}$ $$\mathbf{y} = \begin{bmatrix} 100\\100\\\vdots\\100\end{bmatrix}$$
I am trying to run OLS on this ...
0
votes
0answers
32 views
Similarity matrices and their positive definiteness
Say I have a similarity function $s$ defined on a set $\mathcal{X}$.
Let $\mathbf{S}$ be the similarity matrix between elements in $\mathcal{X}$, i.e.
$\mathbf{S}_{i,j}=s(x_i,x_j), \quad \forall i,j=...
0
votes
1answer
15 views
Explanation on how a matrix $A$ expressed as a product involving a positive semidefinite matrix $\mathcal{H}$ is also positive semidefinite
Suppose we know that that a Hermitian $n \times n$ matrix $A$ can be expressed as the following matrix product
$$A = \begin{bmatrix}
z_1 & 0 & ... & 0 \\
0 & z_2 & ... &...
2
votes
2answers
50 views
How to prove that a function is non-negative definite?
The function I am trying to prove is $\exp(-2\lvert j-k\rvert)$.
Here is what I have tried;
$\sum_{j=1}^n \sum_{k=1}^n\ a_j \bar{a_k}\exp(-2\lvert j-k\rvert)$
=$\sum_{j=1}^n \sum_{k=1}^n\ a_j \bar{...
4
votes
1answer
161 views
Differentiability of the Schatten $p$-norm on positive definite matrices
Let $V$ be the vector space of symmetric matrices in $\Bbb R^{n\times n}$. For $p\in (1,\infty)$, the Schatten $p$-norm of $M\in V$ is defined as $\|M\|_p =(\sum_{i=1}^n \sigma_i(M)^p)^{1/p}$ where $\...
0
votes
1answer
40 views
Sufficient conditions for Loewner ordering of two matrices?
Le $\mathbf{A}$ and $\mathbf{B}$ two psd matrices of size $n$. Additionally we assume that the entries are real and non-negative. Does the following hold:
$$\forall (i,j) \in [n], \mathbf{A}_{ij} \leq ...
0
votes
0answers
17 views
Product of a positive semi definite matrix with an indefinite matrix
I see from my examples that the product of a positive semidefinite matrix(graph Laplacian) and an indefinite matrix (real matrices), comes to be a positive semidefinite matrix. Is there a proof for ...
2
votes
1answer
21 views
Intuitions about positive definite functions
I am looking for further intuitions about positive definite functions, and have several related questions on this matter. I know this isn't the most specific question, but I find that speaking ...
2
votes
0answers
42 views
Minimum of $\langle Ax,x \rangle - 2 \langle b,x \rangle$.
In exercise 5 of section 0 of Fundamentals of convex analysis by Hiriart-Urrut, Lemaréchal, we're supposed to prove that if a self-adjoint linear operator $A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is ...
0
votes
0answers
18 views
Inverse of $(rP+\bar{r}\bar{P})$ where $P=P^*$ and $r=\exp(i\theta)$
I have a positive definite Hermitian matrix $P=P^*>0$ where $P^*$ is the conjugate transpose of $P$ and $r=\exp(i\theta)$. So, how can I prove that
$$ (rP+\bar{r}\bar{P})^{-1}=rY+\bar{r}\bar{Y}$$
...
0
votes
1answer
31 views
Positive definite matrix properties
I am having trouble solving a property that I found.
If $A:n \times n$ is defined as a positive definite matrix and $B: n \times m$ where $rank(B) = r$.
Then $B^T A B > 0$, only when r = m and $B^...
1
vote
0answers
22 views
Determinant of Hadamard product / sum of matrices (one diagonal)
I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $...
0
votes
1answer
16 views
Clarification wrt proof for linear regression cost function being convex
I have trouble understanding the proof which computes hessian of J to see if the optimisation problem is convex.
why is the least square cost function for linear regression convex
The proof claims ...
2
votes
1answer
57 views
showing that map $f$ is positive
Prove that $f: x \to Tx$ is positive on $\mathbb{C}^n$ iff $T$ has ony non-negative eigen values, for a complex $n\times n$ Hermitian matrix $T$.
To prove that $f$ is positive I need to show that $\...
1
vote
1answer
41 views
The positive semi-definiteness of the element-wise matrix product
I understand that if two matrices are PSD, then the element-wise product of the two matrices is also PSD.
However if a matrix in the form $K = A \odot B$ is PSD for any PSD matrix $A$. How about $B$? ...
4
votes
2answers
76 views
What is the Hessian of the spectral norm?
The spectral norm of a symmetric matrix is the absolute value of the top eigenvalue. The gradient of this norm is $uu^T$ where $u$ is the eigenvector associated with that top eigenvalue.
Assume that $...
1
vote
3answers
40 views
How to show that this matrix is positive semidefinite?
Using the definition, show that the following matrix is positive semidefinite.
$$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$
In other words, if ...
1
vote
1answer
31 views
Optimizing over vector and Matrix at the same time.
I want to know if my understand (prove convexity, format as SDP) the following problem is correct:
\begin{equation*}
\begin{aligned}
& \min_{c\in \mathbb{C}^n,D\in \mathbb{H}_+^n} && \|c\|...
1
vote
1answer
47 views
Largest eigenvalue of matrix product $A^T B A$
With $A \in \mathbb{S}^{d \times d}_+$ (symmetric positive semi definite) and $B \in \mathbb{S}^{d \times d}_{++}$ (symmetric positive definite), can we rewrite or upper bound $\lambda_{max}(A^T B A)$ ...
1
vote
0answers
14 views
Equivalence of semidefinite decomposition?
If we have an $n \times n$ positive semidefinite matrix $A$ and we have two decompositions such that
$A = B B^T = C C^T$ for some $n \times n$ matrices $B$ and $C$.
Is it true that $B$ and $C$ are ...
0
votes
1answer
46 views
On what condition $trace(A) \ge trace(AB)$?
Considering $A$ and $B$ are positive semidefinite real symmetric matrices, on what conditions we can have $trace(A) \ge trace(AB)$?
0
votes
1answer
38 views
Can i prove that this matrix is PSD?
I have matrix $A \in \mathbb{R}^{N \times N}$, such that $A(i,j)=trace(B_iCB_j), \forall ij$.
Matrices $B_i$ and C are PSD and symmetric with positive entries. Can I prove that $A$ is PSD too?
In ...
0
votes
0answers
23 views
Why can a constraint on a matrix being positive definite be rewritten as the matrix minus the identity being positive semidefinite?
My instructor today mentioned that if we have a constraint that a matrix $A$ is positive definite, then we can rewrite this constraint as $A - I$ is positive semidefinite without this affecting the ...
1
vote
1answer
39 views
Why taking integral of both sides of matrix inequality is allowed?
How to show if $\nabla^2 f(x) \succeq \alpha I$, then the function is $\alpha$-strongly convex?
In my optimization notes I have
$$\nabla^2 f(x) \succeq \alpha I \rightarrow \alpha\text{-strongly ...
0
votes
1answer
33 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
34 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
34 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 $z^T Xz\geq0\forall z$:
$$\begin{...
0
votes
1answer
28 views
Is this matrix positive semidefinite (Symmetric matrix, with particular pattern)
Let's consider a symmetric matrix A.
If for each row, the diagonal entry is equal or larger than the magnitude of any other element, that is
$$a_{ii} \geq |a_{ij}| \quad\text{for all rows } i \text{...
4
votes
1answer
66 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
$$
S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
2
votes
2answers
67 views
The proof of positive semi-definite for a kernel
How to prove the following kernel $K$ over $\mathbb R \times \mathbb R$ is positive semi-definite:
$$K(x_i, x_j) = e^{-\lambda[\sin(x_i - x_j)]^2},$$
where $\lambda > 0$. It looks like the gaussian ...
1
vote
1answer
23 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
14 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
78 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
98 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 ...
0
votes
0answers
11 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
21 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
36 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 ...
