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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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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 $$ \...
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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 $\Theta$...
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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?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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=...
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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 & ... &...
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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{...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$ (rP+\bar{r}\bar{P})^{-1}=rY+\bar{r}\bar{Y}$$ ...
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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^...
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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 $...
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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
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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 $\...
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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 $B$? ...
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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 $...
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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 ...
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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\|...
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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 $\lambda_{max}(A^T B A)$ ...
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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 ...
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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 $trace(A) \ge trace(AB)$?
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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{...
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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{...
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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} ...
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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 ...
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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 ...
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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?
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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 =...
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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 ...
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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 $\...
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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 ...
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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 ...