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Questions tagged [positive-definite]

For questions about positive definite real or complex matrices. For questions about positive semi-definite matrices, use the (positive-semidefinite) tag.

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I want to check the positive definiteness of the matrix $\Lambda$

Background We consider independent and identically distributed (i.i.d.) random variables $$X_1,\ldots, X_n \overset{\text{i.i.d.}}{\sim} N_p(0, \Sigma).$$ Setup Let $\Sigma$ be a $p$th order positive ...
ytnb's user avatar
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singular positive semi-definite matrix in electromagnetism

anyone knows where he drew this conclusion from?
user900476's user avatar
2 votes
4 answers
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Verify that a quadratic form is NOT positive definite

Verify that the quadratic form $$q(x_1,x_2,x_3)=x_1^2+4x_1x_2+3x_2^2+2x_2x_3+6x_3^2$$ is NOT positive definite and find a vector in $v\in\mathbb{R}^3$ such that $q(v)<0$ . I have made several ...
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1 answer
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Coefficients of the characteristic polynomial and positive definite matrices

I revisited my old notes and saw that my former tutor once told us in Linear Algebra that if we want to check if a matrix $\bf A$ is positive definite, then we can check the coefficients of the ...
metamathics's user avatar
-1 votes
1 answer
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Handling the positive square root of the commutator of two positive definite matrices. [closed]

Let $A,B>0$ be non-commuting positive definite operators. Define the commutator to be $$[A,B] = AB - BA$$ For every operator $A$, $A^*A$ is always positive, and its unique positive square root is ...
Blaine DuBois's user avatar
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60 views

linear algebra - prove that a matrix is positive definite [duplicate]

I'm given a matrix $K$ that is symmetric and positive definite. I'm asked to prove that $K_2 - (l\times u)$ is also a positive definite matrix. $K_2$ is a submatrix of $K$ so that we get rid of the ...
Johann Carl Friedrich Gauß's user avatar
1 vote
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35 views

Proof of Conjecture on `Block-Orthogonalisation Always Reduces Trace'

I have a symmetric positive definite matrix, ${\bf D}\in\mathbb{R}^{NK\times NK}$, which is made up of $K$, $N\times N$, blocks: \begin{equation} {\bf D} = \begin{bmatrix}{\bf D_{11}} & {\bf D_{12}...
Will Dorrell's user avatar
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Prove symmetric matrix $A$ is congruent to $A ^2$ iff $A$ is PSD [duplicate]

How can I solve the following question: Let ( A ) be a symmetric matrix. Prove that ( A ) is congruent to ( A^2 ) if and only if ( A ) is positive semidefinite (PSD) I 've no idea where to start ...
David's user avatar
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How to verify positive definitiveness of the given Kinetic term?

I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12: $$\frac{\mathcal{S}}{\mathcal{T}}= \int dt\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)+11.3c_0^...
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Smallest eigenvalue of $A^T D^{-1} A + D$ for positive diagonal $D$.

I am wondering whether or not there exists a way to analyze the eigenvalues of $$A^T D^{-1} A + D$$ for a square matrix $A \in \mathbb{R}^{n \times n}$ and positive diagonal matrix $D\in \mathbb{R}^{n ...
Vito Zarra's user avatar
6 votes
1 answer
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$\int_{x_i>0} f(x_1+x_2, x_3+x_4)^* f(x_1+x_3, x_2+x_4) \;(x_1x_2x_3x_4)^\alpha \geq0$ for any function $f$ and $\alpha >-1$?

Let $f(u,v)$ be a "nice" complex-valued function defined on the region $u,v\geq 0$, e.g., $f$ is continuous and decays rapidly at infinity. Problem: For any such function $f$ and for any $\...
Laplacian's user avatar
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Proving positive semidefiniteness of a matrix multiplication

Assume $Q \in \mathbb R^{n\times n}$ is positive semidefinite and $I$ is the identity matrix. How can I prove that the matrix $$M = (I + \alpha Q)^{-1} Q $$ is positive semidefinite for all $\alpha \...
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Definiteness of quadratic form with block matrix structure

Consider the quadratic form: $$f(x,y) = \sum_{i}c_{i}x_{i}y_{i} + \sum_{ij}(A^{-1})_{ij}x_{i}x_{j} + \sum_{ij}(B^{-1})_{ij}y_{i}y_{j}$$ where $A,B\in\mathbb{R}^{n\times n}$ are symmetric and positive ...
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Showing positive definiteness of joint process with cross-correlation

This paper claims that if $Y_1$ and $Y_2$ are processes whose covariance matrices are $$\mathsf{Cov}(Y_1)=L_1 L_1^\top$$ $$\mathsf{Cov}(Y_2)=L_2 L_2^\top,$$ where $L_1$ and $L_2$ are square root ...
Remy's user avatar
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$\int_{x_i>0} f(x_1+x_2, x_3+x_4)^* f(x_1+x_3, x_2+x_4) \geq0$ for any function $f$?

Let $f(u,v)$ be a "nice" complex-valued function defined on the region $u,v\geq 0$, e.g., $f$ is continuous and decays rapidly at infinity. Problem: For any such function $f$, prove that $$\...
Laplacian's user avatar
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Can we prove that ABA' is positive definite if B is positive definite and A is of full row rank

Suppose $A$ is a $r \times n$ matrix and $Rank(A) = r$, $B$ is a $n \times n$ symmetric matrix and $rank(B) = n$. Can we prove that $ABA'$ is positive definite? If it is not positive definiteness, ...
shani's user avatar
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Maximal value of $\sum_{n=1}^N z^T X(P_n X + I)^{-1}z$ over positive semidefinite cone?

Let $z$ denote a unit vector. Fix a finite collection of positive semidefinite matrices $\mathcal{P}$. Define the function and set $$ f_{\mathcal{P}}(X) = \sum_{P \in \mathcal{P}} z^T X(PX + I)^{-1} z,...
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Why is the function $x\mapsto |x|^a$ ($a\in ]0,2]$) negative definite?

I'm studying negative definite functions from the book C. Berg, and G. Forst, Potential Theory on Locally Compact Abelian Groups (Springer Berlin Heidelberg, Berlin, Heidelberg, 1975). In this book, a ...
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Determinant upper bound for specific matrix form

Let $A \in \mathbb{R}^{d \times d}$ be a symmetric matrix with the following properties: $A_{ij} \leq 0 \text{ for } i \neq j$ $A_{ii} \geq 0 \ \forall i \in [d]$ $\sum\limits_{i=1}^d A_{ij} = 0 \ \...
guysc's user avatar
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Suppose $A,B\in M_n(\mathbb{R})$ are symmetric, $A-B$ is semi-positive-definite, and $A$ has the same eigenvalues as $B,$ can we derive that $A=B$?

Let $A,B$ be two real symmetric matrices, denote their eigenvalues as $\lambda_1\geq\cdots\geq\lambda_n$ and $\mu_1\geq\cdots\geq\mu_n$ respectively. Suppose $A-B$ is semi-positivedefinite and $\...
Tiffany's user avatar
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2 votes
1 answer
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Trace of symmetric part of product of 3 positive semidefinite matrices

Suppose we have 3 symmetric positive semi-definite matrices $A,B,C$. How can one prove or give counter-example for the following statement? $$ \mathrm{trace}(AB(A-C)^2 + (A-C)^2BA) \ge 0 $$ where we ...
sixtyTonneAngel's user avatar
1 vote
0 answers
38 views

Definiteness of an integral quadratic form

Given $A:\mathbb R^n \rightarrow \mathbb R^{n\times n}$, $x\mapsto A(x)$ with eig$\big(A(x)\big)\subset \mathbb C^-:= \{\lambda \in \mathbb C \vert Re(\lambda) <0\}$ for all $x$. Given also $\...
Name123's user avatar
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Degenerate spectrum for the sum of projections onto unit vectors that sum to zero implies a symmetry of the vectors?

I am interested in a sum of projections $A = \sum_{i=1}^N \mathbf{a}_i\,\mathbf{a}_i^T$, where $\mathbf{a}_i$ are real column vectors with unit norm and $\sum_i \mathbf{a}_i = \mathbf{0}$. So $A$ is a ...
Luzveraz's user avatar
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lower bound on the eigenvalues of a particular PD matrix

Consider the matrix $A = \begin{bmatrix} A_1 &... & A_m\end{bmatrix} \in \mathbb{R}^{n \times m}$ with $m > n$. Assume $A$ has full rank. Then let $D = diag \{ \lambda_1, ..., \lambda_m\}$ ...
C Marius's user avatar
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1 vote
1 answer
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Conditions for a Hermitian matrix to be positive definite

It's a question from chapter 7c of Linear algebra done right 4th edition. Suppose $T \in \mathcal{L}(V)$ and $e_1 \cdots e_n$ is an orthonormal basis of $V$. Prove that T is a positive operator if ...
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1 answer
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Is this function positive (semi)-definite?

Let $A$ be a real positive (semi) definite matrix and $$\varphi: \mathbb R^d \to \mathbb R$$ with $$ \varphi(\mathbf x) = \mathbf x^T A \mathbf x $$ Is $\varphi$ positive (semi) definite? For ...
TRP's user avatar
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1 vote
1 answer
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Positive definiteness of a diagonal matrix and boundedness of specific set

Problem: Let $D\in \mathbb R^{n\times n}$ be a diagonal matrix. Show that the set $$E=\{x\in \mathbb R^{n\times 1} \mid x^tDx\leq 1 \}$$ is bounded if and only if the diagonal entries of $D$ are ...
categoricallystupid's user avatar
2 votes
0 answers
54 views

Positive definite matrices and pivots.

I found these lecture notes online about positive definite matrices. To summarise what's needed for my question, they give 4 ways of working out if a matrix is positive definite. In particular, it ...
Mathematista's user avatar
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0 answers
29 views

Effect of row operations on the sign of eigenvalues/ positive definiteness of remaining submatrix

Suppose, we have a real, symmetric, positive definite matrix $M$. We know that M is positive definite if and only if all of its eigenvalues are positive. My question is whether (Gaussian) eliminating ...
cbakos's user avatar
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0 answers
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Proving inequality involving mean, covariance and their estimate.

Let $$ A = \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right) (\mu - \hat \mu) + \gamma \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right)(\hat \Sigma \iota), \...
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1 answer
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Proving inequality regarding expected error involving covariance and its estimate.

Let $$ A = \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right)(\hat \Sigma \iota).$$ Where $\Sigma$ is the true $nxn$ covariance matrix of a random vector x that is normally ...
alejandroll10's user avatar
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0 answers
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Positive definiteness of the derivative of a real-valued positive definite matrix

Is the derivative of a real-valued positive definite matrix is also a real-valued positive definite matrix? If that is not always the case, when it is guaranteed?
Omar Shehab's user avatar
1 vote
1 answer
54 views

Making a symmetric matrix positive (semi-)definite by adding a diagonal matrix

Say I have a matrix $A \in R^{p \times p}$ which is symmetric and with non-negative diagonal entries (i.e. $a_{ii} \geq 0 \forall i\in \{1, \ldots, p\}$). However, $A$ is not positive (semi-)definite. ...
HeyCool08's user avatar
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1 answer
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How to show the quadratic form of the following matrix converges to zero?

Suppose we have a $l\times l$ real matrix $M=(X'X)^{-1}X'\Sigma X (X'X)^{-1}$, where $X$ is a $n\times l$ real matrix and $\Sigma$ is a $l\times l$ real symmetric and positive definite matrix. $X'X$ ...
ExcitedSnail's user avatar
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0 answers
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Solving a "quadratic equation" in $\mathbb{R}^n$ with linear constraints on $\mathbb{R}^n$

Problem: Given that $L$ is a subspace of $\mathbb{R}^n$ and denote $L^\perp$ as the orthogonal complement of $L$, i.e. we can write $\mathbb{R}^n = L \bigotimes L^\perp$. Let $w \in \mathbb{R}^n$ be a ...
ElementX's user avatar
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2 votes
0 answers
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subset of positive matrices such that the Loewner order is total

Are there known characterizations of subsets of positive matrices over which the Loewner order is total ?
yosh's user avatar
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0 answers
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Would $v^TAv>0$ forall $v$, $v$ if A's eigenvalues are all > 0? [duplicate]

I'm learning that if $n \times n$ positive definite matrix A A's then $\forall v$ $v^TAv>0 \ $, but this requires A is symmetric(then A have orthonormal eigenvectors) so I'm asking: 1.if A is not ...
femto's user avatar
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0 answers
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Show that to reduce the initial error by the factor $\varepsilon$ at most many iteration steps are required

Consider the CG method for the iterative solution of the system of equations $A \mathrm{x}=\mathrm{b}$ with a positive definite and symmetric matrix $A \in \mathbb{R}^{n \times n}$ and $b \in \...
Euler007's user avatar
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1 answer
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Show in terms of numerics for symmetric and positive definite matrix

Let $0<\lambda_{1} \leq \ldots \leq \lambda_{n}$ be the eigenvalues of the symmetric and positive definite matrix $A \in \mathbb{R}^{n \times > n}$ and $u_{1}, \ldots, u_{n} \in \mathbb{R}^{n}$ ...
Euler007's user avatar
  • 132
1 vote
1 answer
22 views

For a fixed positive definite $A$, is $(x,B) \to x^TB^{-1/2} A B^{-1/2} x$ always jointly convex?

Consider a real positive definite matrix $A\in R^{d\times d}$. Let $S_d^+$ be the set of symmetric positive matrices. Is it always true that $$(x,B) \to x^TB^{-1/2} A B^{-1/2} x$$ is jointly convex in ...
jlewk's user avatar
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0 answers
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Infer the positive definiteness of a real non-symmetric matrix from the positive definiteness of its symmetric part

Let $A$ be a non-symmetric matrix of real numbers, $A\in\mathbb{R}^{n\times n}$. If the quadratic form $x^\intercal Ax$ is positive definite in the sense that $x^\intercal Ax>0$ $\forall x\neq 0\in\...
fma's user avatar
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0 votes
1 answer
23 views

Positive Matrices and Linear Forms

Fix a vector $\vec{b}$ and a positive definite (but not necessarily symmetric) matrix $A$, can we prove that the fraction $$ \frac{\vec{b}^TA\vec{y}}{\vec{b}^T\vec{y}} $$ always has the same sign (If $...
TheWildCat's user avatar
1 vote
0 answers
50 views

Given matrix A with $A_{i,j} = f(i,j)$, how can I find out if A is positive semidefinite?

I'm working on a problem whose solution relies on finding if an arbitrary matrix A is positive semi-definite. A is real valued, square and symmetric, and each of its entries are given by: $A_{ii} = f(...
William's user avatar
  • 147
0 votes
1 answer
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Sum of positive semi-definite matrix and positive definite matrix?

Is the sum of a positive semidefinite matrix and positive definite matrix a positive definite matrix? I have a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$ and the identity $I_n$, is ...
atul ganju's user avatar
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0 answers
22 views

how to show that a nonsingular positive semidefinite matrix is positive definite?

Got a question regarding the following statement: a nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how to show it. If the above statement is not true, how about ...
ExcitedSnail's user avatar
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0 answers
13 views

Factorizing $AMA^T+N=WW^T$ efficiently.

This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed. ...
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2 answers
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If $A$ is a symmetric positive definite matrix, show that $f(x) = x^TAx$ is convex.

Let: \begin{gather*} f: \mathbb{R}^n \to \mathbb{R}, \quad A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n, x \in \mathbb{R}^n, c \in \mathbb{R} \\ f(x) = x^T A x + b^T x + c \\ \end{gather*} If ...
clay's user avatar
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1 vote
2 answers
46 views

Sum of norms induced by PSD matrices

Suppose you have two positive definite matrices $M$ and $N \in \mathcal{S}_{++}^n$. They induce scalar products and norms on $\mathbb{R}^n$: \begin{align} \lVert x \rVert_M &= \sqrt{x^\intercal ...
Tasty's user avatar
  • 102
0 votes
1 answer
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$0 ≤ A ≤ I. $ $\iff$ ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$

Given two operators $A $and$ B$, where $A ≤ B$ means the operator $B − A$ is positive semidefnite. (i) $0 ≤ A ≤ I.$ (ii) ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$ are equivalent I am having ...
darkside's user avatar
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0 answers
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For an $n$ by $n$ symmetric matrix $M$, why does $Tr(M)^2/Tr(M^2)>n-1$ imply that M is positive or negative definite?

Let $M$ be an symmetric square matrix of size $n$. I am trying to prove that $Tr(M)^2/Tr(M^2)>n-1$ is a sufficient condition for proving that $M$ is positive or negative definite. If in addition $...
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