Questions tagged [positive-semidefinite]
Relating to a symmetric $n\times n $ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$
1
vote
1answer
13 views
Proof of why $\mathbf{A}_{ij}=i(n-j+1)$ for $j \geq i$ indicates positive semidefiniteness
Problem
When reading an advanced text in numerical computing, I encountered the following claim
If a symmetric matrix $\mathbf{A}\in \mathbb{R}^{n\times n}$ has entries $\mathbf{A}_{ij}=i(n-j+1)$ ...
2
votes
0answers
26 views
How to show for positive semi-definite, there exist $x \in \mathbb{R}^n$ such that $Bx+c = 0$ and and $\|x\| \leq \Delta$?
Let $B \in \mathbb{R}^{n \times n}$ be symmetric and positive semi-definite such that $B = U\Lambda U^T$, where $U = [u_1,\cdots,u_n]^T$ is an orthogonal matrix with $u_i \in \mathbb{R}^n$, and $\...
0
votes
1answer
38 views
Inequality w.r.t determinant of a non-negative definite matrix.
I am reading a paper where the author mentioned the following property without proof. Neither can I prove it nor can I find the proof in various textbooks.
For any non-negative definite (i.e. ...
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votes
2answers
33 views
Numerically verify a matrix is positive semidefinite
I am trying to numerically (in Julia) verify that
A symmetric matrix $\mathbf{A}$ is positive semidefinite if and only if it is a covariance matrix.
Then I need to verify in both directions, i.e.
...
0
votes
1answer
18 views
Positive trace (all diagonal entries are positive) implies semipositive definite?
I am working on a matrix with all the diagonal entries are strictly positive while every other entry is strictly negative. This matrix is symmetric as well.
I want to show that this matrix is ...
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0answers
20 views
Monotonicity of matrix inverse of positive definite matrices [duplicate]
If $A$ and $B$ are positive definite, $A-B$ is positive definite, can we say $B^{-1}-A^{-1}$ is positive definite? I think this should be true but I don't know how to prove it.
It would be great if ...
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0answers
47 views
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 ...
2
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0answers
62 views
Semidefinite optimization
I'm a physicist and I'm working on a problem that can be reduced to a SDP problem. My problem is: is there a theorem that assures that the result of an optimization saturates the constraint instead of ...
0
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0answers
37 views
Find a counter example to majorization theory
Theorem: Let $X$ be an $n\times n$ positive semi-definite Hermitian matrix with diagonal elements $d=[d_1\cdots d_n]'$ and eigenvalues $\lambda=[\lambda_1\cdots\lambda_n]'$, where $d_1\geq\cdots\geq ...
0
votes
1answer
33 views
Positivity of a matrix
Let $A$ be a $3\times 3$ matrix defined in the following way:
$$A=\begin{bmatrix} a & c & 0\\c & b &-c\\0 & -c & 1-a-b\end{bmatrix}$$
I wish to show that $A=BB^t$ for some ...
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0answers
25 views
Proving that the difference of two matrices is positive semi-definite
I would like some hint to show that the following matrix is positive semi-definite. Thanks for any help in advance.
$(\text{X}^{T} \Omega^{-1}\text{X})^{-1}\text{X}^{T}\Omega^{-1}\Omega^{-1}\text{X}(\...
-2
votes
1answer
36 views
How to simplify this trace term?
I have the following trace term:
trace(Sk' Ck Sk)
where Sk is a KxM matrix and Ck is a KxK positive semidefinite matrix.
I'm involving this trace term in a ...
1
vote
1answer
60 views
Determinant of a positive semi-definite matrix
If $M$ is a Hermitian matrix, then $M$ is positive definite if and only if its leading principal minors have positive determinant, i.e the following matrices have positive determinant:
The upper-left ...
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0answers
30 views
Can eigenvalues of a correlation matrix be negative?
Can eigenvalues of a correlation matrix be negative?
I am a bit confused because I know that a correlatin matrix $C$ is always positive semi-definite ie for all $v \in \mathbb R^n$ we have $v^T C v \...
2
votes
1answer
61 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
$$ \...
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0answers
13 views
Absolute convex hull of rank 1-correlation matrices?
Does there exist a ''universal'' constant, $c > 0$ say, such that for any(!) $k \in \mathbb{N}$ every(!) $k \times k$-correlation matrix $\Sigma$ can be written as $\Sigma = c\Theta$, where $\Theta$...
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vote
2answers
135 views
Is a square zero matrix positive semidefinite?
Does the fact that a square zero matrix contains non-negative eigenvalues (zeros) make it proper to say it is positive semidefinite?
2
votes
2answers
45 views
Proving the difference of two matrices is PSD
Claim: For $x\in \mathbb{R}^n$, we have $\operatorname{Diag}(x) - xx^T \succeq 0$ if and only if $x_i \geq 0 \ \forall i\in [n]$ and $\sum_{i} x_i \leq 1$.
Where $\operatorname{Diag}$ denotes the ...
1
vote
1answer
14 views
Negating off-diagonal blocks retains positive-semidefiniteness?
I am trying to follow some notes that state
$$
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$$
and ...
0
votes
1answer
27 views
Nearest positive semi definite matrix to a complex valued Hermitian matrix
How would you find the nearest (via Hilbert distance), PSD matrix (with trace = 1) to a Hermitian matrix? I found an answer to a similar question here. However, as I understand, Hingham's work only ...
2
votes
2answers
44 views
If both $A-B$ and $B-A$ are positive semidefinite, then $A = B$
Let $A, B$ be two positive semidefinite matrices. Prove that if both $A-B$ and $B-A$ are positive semidefinite, then $A = B$.
I can show that their diagonal elements are the same but for others I ...
0
votes
1answer
116 views
How to prove the logistic loss function is strongly convex?
The logistic loss function is:
$$\mathcal{L}=\frac{1}{n}\sum_{i=1}^n\log(1+\exp(-y_ix_i^T\theta))$$
in which $y_i\in\{-1,+1\},x\in \mathbb{R}^d$. How to show that $\mathcal{L}$ is strongly convex.
My ...
3
votes
1answer
59 views
Matrices Inequality Proof
Recently, I read a paper and there is a step which turns out not obvious to me. The statement is as follows:
All matrices here are real matrices. $F$ is an arbitrary square matrix. $\Psi$ is a ...
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votes
0answers
32 views
Non-Negative Vs Positive Semi Definite
A matrix is PSD if $$\langle Ax, x\rangle \ge 0, \forall x \in H$$
Where, H is a hilbert space and A is a mapping $H \rightarrow H$. Is it the same as being Non-negative? I couldn't seem to find a ...
2
votes
2answers
94 views
Parametrization of unitary matrices
Does anyone know a simple way to parametrize the space of $n\times n$ complex unitary matrices into a set of independent complex numbers in some complex-rectangle? Specifically the mapping and inverse ...
1
vote
0answers
12 views
Semidefinite Matrix in LINGO
Using LINGO, I need to enter the following block matrix as one of my constraints
$
M=
\left[ {\begin{array}{cc}
1 & x^T \\
x & X \\
\end{array} } \right]
$
where x is an n by 1 ...
0
votes
2answers
33 views
Sum of symmetric, positive semidefinite matrices
Let $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{p \times n}$. Show that $A^{T}A+ B^{T}B$ is invertible if and only if $\ker A \cap \ker B =\lbrace 0 \rbrace$.
I could show that if it's ...
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votes
0answers
28 views
Why would sequential quadratic programming fail to find global minimum?
I have a data set. A matrix $X$, $1300 \times 20$ and output vector $\mathbf{y} \in \Bbb R^{20}$ $$\mathbf{y} = \begin{bmatrix} 100\\100\\\vdots\\100\end{bmatrix}$$
I am trying to run OLS on this ...
0
votes
1answer
15 views
Explanation on how a matrix $A$ expressed as a product involving a positive semidefinite matrix $\mathcal{H}$ is also positive semidefinite
Suppose we know that that a Hermitian $n \times n$ matrix $A$ can be expressed as the following matrix product
$$A = \begin{bmatrix}
z_1 & 0 & ... & 0 \\
0 & z_2 & ... &...
2
votes
2answers
75 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
177 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
45 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:
$$\forall (i,j) \in [n], \mathbf{A}_{ij} \leq ...
0
votes
0answers
19 views
Product of a positive semi definite matrix with an indefinite matrix
I see from my examples that the product of a positive semidefinite matrix(graph Laplacian) and an indefinite matrix (real matrices), comes to be a positive semidefinite matrix. Is there a proof for ...
2
votes
1answer
29 views
Intuitions about positive definite functions
I am looking for further intuitions about positive definite functions, and have several related questions on this matter. I know this isn't the most specific question, but I find that speaking ...
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votes
0answers
42 views
Minimum of $\langle Ax,x \rangle - 2 \langle b,x \rangle$.
In exercise 5 of section 0 of Fundamentals of convex analysis by Hiriart-Urrut, Lemaréchal, we're supposed to prove that if a self-adjoint linear operator $A:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is ...
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votes
0answers
18 views
Inverse of $(rP+\bar{r}\bar{P})$ where $P=P^*$ and $r=\exp(i\theta)$
I have a positive definite Hermitian matrix $P=P^*>0$ where $P^*$ is the conjugate transpose of $P$ and $r=\exp(i\theta)$. So, how can I prove that
$$ (rP+\bar{r}\bar{P})^{-1}=rY+\bar{r}\bar{Y}$$
...
0
votes
1answer
32 views
Positive definite matrix properties
I am having trouble solving a property that I found.
If $A:n \times n$ is defined as a positive definite matrix and $B: n \times m$ where $rank(B) = r$.
Then $B^T A B > 0$, only when r = m and $B^...
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vote
0answers
51 views
Determinant of Hadamard product / sum of matrices (one diagonal)
I am trying to compute the determinant of $\boldsymbol{W}\odot \boldsymbol{S}$, where $\boldsymbol{S} \in PD(p)$ positive semidefinite matrix and $\boldsymbol{W}$ is a matrix whose diagonal entries $...
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votes
1answer
28 views
Clarification wrt proof for linear regression cost function being convex
I have trouble understanding the proof which computes hessian of J to see if the optimisation problem is convex.
why is the least square cost function for linear regression convex
The proof claims ...
2
votes
1answer
58 views
showing that map $f$ is positive
Prove that $f: x \to Tx$ is positive on $\mathbb{C}^n$ iff $T$ has ony non-negative eigen values, for a complex $n\times n$ Hermitian matrix $T$.
To prove that $f$ is positive I need to show that $\...
2
votes
1answer
42 views
The positive semi-definiteness of the element-wise matrix product
I understand that if two matrices are PSD, then the element-wise product of the two matrices is also PSD.
However if a matrix in the form $K = A \odot B$ is PSD for any PSD matrix $A$. How about $B$? ...
5
votes
2answers
86 views
What is the Hessian of the spectral norm?
The spectral norm of a symmetric matrix is the absolute value of the top eigenvalue. The gradient of this norm is $uu^T$ where $u$ is the eigenvector associated with that top eigenvalue.
Assume that $...
1
vote
4answers
56 views
How to show that this matrix is positive semidefinite?
Using the definition, show that the following matrix is positive semidefinite.
$$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$
In other words, if ...
2
votes
1answer
35 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
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1answer
54 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 ...
1
vote
1answer
50 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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votes
1answer
41 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
25 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
40 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 ...
