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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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Weighted inner product with arbitrary matrix?

An inner product can be written in Hermitian form $$ \langle x,y \rangle = y^*Mx $$ that requires $M$ to be a Hermitian positive definite matrix. I have read that using Hermitian positive definite ...
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Degrees of freedom of the set of positive definitive matrices

If I am not wrong, the set of definite positive matrices with real coefficients is a convex cone without the vertex, which is the null matrix. What is the number of degrees of freedom for this set of ...
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Is this matrix defined from an integral of a non-negative function positive definite?

I have a function $f(x)$ defined as \begin{equation} f(x) := \sum_{n = 1}^N c_n f_n(x) = \mathbf{f}(x)^T \mathbf{c}, \end{equation} where \begin{align} \mathbf{f}(x) := \begin{bmatrix} f_1(x) \\ ...
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1answer
10 views

using one vector to build positive definite matrix

Let $X=(X_1,X_2,...,X_n)_{1\times n}$ be a n-dimensional vector and a matrix $A_{n\times n}=(X^{T})_{n\times 1} * X_{1\times n}$. Under what condition of $X$, $A$ is a (semi-)positive definite matrix? ...
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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
35 views

Inequality for trace of product of matrices

Assume that $A \in \mathbb{R}^{n \times n}$ is a symmetric matrix and $B \in \mathbb{R}^{n \times n}$ is a symmetric positive definite matrix. Is the following statement true $$ \lambda_{\mathrm{min}} ...
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1answer
30 views

How to prove that $\langle P,A^2 \rangle \le 0 $ for every positive $P$ and skew-symmetric $A$?

I have stumbled upon the following claim, and I wonder if it has a simple proof: Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\...
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Approximation of a positive definite matrix

I have a covariance matrix (A), which is positive definite. I would like to approximate matrix A by another positive definite matrix B in such a way, that the eigenvalues of B span only 2 orders of ...
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43 views

Principal Minor Theorem for dimension 2

Let $A$ be a real ($2 \times 2$)-matrix such that the map $\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$, $\langle x,y \rangle = x^t A y$ is a scalar product. Now I would like to show that the ...
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45 views

how to prove the following linear matrix inequality

$K=\{1,2,\dots,m\}$ $A_1,\dots, A_m$ be any square matrices. $n\times n$ say. If there exist positive definite symmetric matrices $P_1,\dots, P_m$, $(n\times n)$ say. such that, $A_i^TP_jA_i-P_i<...
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Compare ratio of two determinants before and after adding sum of rank 1 matrices in both determinants

I have a ratio between two determinants: $$r_1 = \frac{\det(\lambda I_d + \sum_{t=1}^{n}x_{t}{x_{t}}^T)}{\det(\lambda I_d)}$$ where $x \in \mathbb{R}^{d}$, $||x||_{2} \leq 1$, $\lambda > 0$, and $...
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Cholesky decomposition: Check if $A$ is positive definite

How can we show that a matrix $A$ is not a positive definite matrix using the Cholesky decomposition? If we are not able to complete the algorithm and we cannot factor the matrix with a Cholesky ...
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1answer
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Defining Hermitian Adjoints Non-degenerate Hermitian Forms that are NOT positive definite.

I was looking around different textbooks and websites for the definition of a Hermitian adjoint. All the resources that I have checked including the one I am studying at the moment (Jeevanjee's Intro. ...
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1answer
66 views

Positive Definiteness of Arcsin of a Positve Definite Matrix [closed]

Suppose that $M$ is a positive definite matrix with entries within $[-1,1]$, and let $N$ be a matrix where $N_{ij} = \sin^{-1}M_{ij}$. How do I show that $N$ is also positive definite?
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Conditionally (almost) definite matrices and Hadamard product

There seem to be different uses of the terminology. One says that a matrix $M$ is almost definite if $x'Mx=0 \Rightarrow Mx=0$. But here I am referring to a different one, that is there exist some $A$ ...
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1answer
42 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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1answer
66 views

Show a polynomial of degree 4 in 7 variables is positive

I am trying to find out if the following homogeneous polynomial of degree 4 in 7 variables ($a,b,c,d,x,y$ and $z$) is positive (except when all variables are zero). After running numerical ...
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1answer
58 views

Fourier Transform of $e^{-<x,Ax>}$, $A$ is a symmetric, positive definite matrix

I would like to understand how to fourier transform the function $f:\mathbb R^n\to\mathbb R$, $f(x):=e^{-\lt x,Ax\gt}$ with $A$ being a positive definite, symmetric matrix. I understand that $\lt x,...
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2answers
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How to show $\text{Tr}(M\log N)=\sum_{i,j}^n\lambda_i\log(\tilde{\lambda_j})(u_i^{\top}\tilde{u}_j)^2$?

The above question is the equation $(2.4)$ of the following paper: MATRIX EXPONENTIATED GRADIENT UPDATES. Let $M$ and $N$ be two $n \times n$ positive definite matrices where $M=U\Lambda U^{\top}$, $...
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Condition for positive definiteness of matrix difference

Suppose I have two matrices $X \in \mathbb{R}^{n \times q}$ and $Y \in \mathbb{R}^{n \times q}$. Suppose that both $X$ and $X - Y$ have rank $q$ but $Y \neq 0$. I want to determine if there are any ...
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Representation of negative Quantum entropy in terms of eigenvalues, i.e., $\text{Tr}(M\log M -M)=\sum_{i=1}^{n}(\lambda_i\log(\lambda_i)-\lambda_i)$?

Negative Quantum entropy or Negative Von Nuemann entropy is defined as $f(M)=\text{Tr}(M\log M -M)$. Where $M$ is a positive definite matrix in $\mathbb{S}_+^n$, $\log$ is natural matrix logarithm ...
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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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On positive definiteness of a sub-matrix after first step Gaussian elimination to a symmetric positive definite matrix

Let $A=[a_{ij}]\in M_n(\mathbb R)$ be a symmetric positive definite matrix (i.e. all eigenvalues of $A$ are real and positive ) with $a_{11}\ne 0$ . Now let $A_1=[a' _{jk}] \in M_{n-1}(\mathbb R)$ ...
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How to show the trace inequality of two P.S.D matrices $\text{Tr(X)}\leq\text{Tr(Y)}$ when $X \preceq Y$?

Let $X,Y$ be two Positive Semi-Definite matrices. How can we show the following in the most elegant and shortest way? Because I know how to prove it but I think there is a better way? Alos, MaoWao ...
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25 views

Definiteness of matrix after Woodbury inversion.

Consider a real, symmetric and positive definite $n\times n$ matrix $\mathbf{K}$, and a $n\times m$ matrix $\mathbf{W}$. $\mathbf{W}$ contains $m$ columns with all zeros except a single entry in each ...
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329 views

Is the following matrix positive definite?

Is there an easy way to prove / disprove this? $$ (X)_{ij} = \begin{cases} \dfrac{k}{n}& \text{if}\ i = j \\ \dfrac{k(k-1)}{n(n-1)} & \text{otherwise} \\ \end{cases} $$ where $X \in \...
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On the entries of $LL^t$ where $L \in GL_n (\mathbb R)$ is lower triangular with positive diagonal entries

Let $L \in GL_n (\mathbb R)$ be a lower triangular matrix with positive diaginal entries and let $A :=LL^t$ . (note that $A$ is positive definite i.e. $A$ is symmetric and all eigenvalues of $A$ are ...
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27 views

Positive Definite Matrices and eigenvalues

Problem: Let $A$ ∈ ${C}^{n×n}$ be, such that for every x ∈ $C^n$ <$A$x$, x$> ≥ 0 Show that all eigenvalues of $A$ are positive or zero I suppose that from the standart inner product in the ...
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1answer
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If A is a symmetric square matrix. I need to show that it is positive definite only if all eigenvalues are positive.

I understand that a positive definite matrix by the definition is a symmetric matrix where all eigenvalues are positive. I also know that if $ (x,y) = {x^T}{\cdotp}M{\cdotp}y$ then it is positive ...
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Linear Algebra - Positive-Definiteness in Vectors?

I was reading up on the inner product over at this Wikipedia page, and I noticed, in the given definition, the use of the term "positive-definiteness". Now, from what I know, this is terminology one ...
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1answer
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Definiteness of a submatrix of a positive definite matrix

Consider a real, symmatric and positive definite $n \times n$ matrix $\mathbf{A}$, and a $n \times m$ matrix $\mathbf{W}$. $\mathbf{W}$ contains $m$ columns with all zeros except a single entry in ...
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Existence of a sub-matrix capturing most of the power of the whole matrix

I would like to verify the following argument: Consider an $M \times N$ complex matrix $\mathbf{X}$ with $M \le N$. There exists a sub-matrix $\tilde{\mathbf{X}}$ containing $M$ columns of $\mathbf{...
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If I have a positive definite matrix X. How do i show that X$^2$ and X$^{-1}$ are also positive definite?

To my understanding a positive definite matrix is a real symmetric square matrix where all eigenvalues are positive. Therefore for a matrix A and vector v $Av = {\lambda}v$ where $\lambda$ is an ...
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58 views

if $A>0$, prove that there exists $x\in\{\pm1\}^n$, such that $(Ax)_i\neq0$ for all $i$

If $A\in\mathbb{R}^{n\times n}, A>0$ (symmetric, positive definite), prove that there exist $x=\begin{bmatrix} \sigma_1 \\ \vdots \\ \sigma_n\end{bmatrix}$, $\sigma_i\in\{\pm1\}$, such that $(Ax)_i\...
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1answer
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Are all symplectic $(0,1)$-matrices lower/upper block-triangular?

Context. I don't expect this question is actually interesting, it just seems like a nice/fun exercise to get better acquainted with $Sp(2n)$ (and to celebrate the new year!). In this question ...
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1answer
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A question about the symmetric positive definite matrix A and $D^{-1/2}AD^{-1/2}$

Assume that $A\in\mathbb{R}^{n\times n}$ is a symmetric positive definite matrix and $D=diag(d_1,\ldots,d_n)$ is a diagonal matrix constructing by using the diagonal entries of $A$, indeed $d_i=a_{ii}$...
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Positive definiteness of a Runge-Kutta method

A Runge-Kutta method can be characterized by the $s \times s$ matrix $A$ and the $s$-element column vectors $\mathbf{b}$ and $\mathbf{c}$. In this paper, a special type of Runge-Kutta method is ...
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44 views

How can we decompose the identity matrix given a set of orthonormal vectors?

Let $A$ be a positive semidefinite (P.S.D) matrix with distinct set of eigenvalues. since it is P.S.D its eigendecomposition is as follows for eigenpairs of $(\lambda_i,v_i)$ $$ A= \begin{bmatrix} ...
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Is there a closed-form formula for the derivative of the positive factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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Is it true that $A^{T}P+PA>0$ for an unstable matrix $A$ and a positive definite $P$?

For a positive definite matrix $P$ and a matrix $A$ with all positive eigenvalues, how to guarantee that the matrix $Q=A^{T}P+PA$ is positive definite? I know if $A$ is a stable matrix (i.e. all the ...
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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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Trace of a product of two positive definite matrices

Let $A, B, C_t \in \mathbb{R}^{n \times n}$ be positive definite matrixand $C_t$ is defined as \begin{align*} C_{t,(i,j)}=\begin{cases} A_{i,j}, &i,j < t \\ B_{i,j}, &i,j \ge t \\ 0, &...
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Confusion about the uniqueness of the square root of a positive definite matrix

There is a lot of material showing that a positive definite matrix $A$ has a unique positive definite square root, $B$ such that $B^2=A$. During a self study session, I needed to use this fact for a ...
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122 views

An inequality for positive definite matrix with trace 1

Given a positive definite matrix $A \in \mathbb R^{n \times n}$. If $\operatorname{trace}(A) = 1$, for $n \geq 3$, prove that $$\text{det}(A) \leq \frac{n^n}{(n-1)^{2n}} \text{det}(I -A)^2$$ and the ...
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Bound of the Subtraction of Two Inverse Matrix

Let $\bf{A, A'_k}\in \mathbb{R}^{NxN}$ be symmetric positive definite. For some $1\le k \le N$, $\bf{A'_k}$ is defined as $$\bf{A'_k} = \begin{pmatrix} \bf{A}_{(1:k-1), (1:k-1)} & 0 \\ 0 & \...
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1answer
51 views

If $B$ is a small perturbation of positive-definite matrix $A$, then do we have $B>\epsilon A$?

Suppose $A=(a_{ij})$ is a (symmetric) positive-definite matrix, and $B$ is another symmetric matrix. Question: If $B$ is in a small neighborhood $U$ of $A$, then it seems that $B$ should also be ...
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2answers
14 views

Positive definite matrix if eigenvalue has positive and negative solutions

If a matrix $A$ is Hermitian and its eigenvalues have positive and negative solutions, is it still considered to be positive definite? For example, is the following matrix positive definite?
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1answer
43 views

Maximize $\det X$, subject to $X_{ii}\leq P_i$, where $X>0$

Given $P_1,P_2,\cdots,P_N$. \begin{array}{ll} \text{maximize} & \det X\\ \text{subject to} & \mathrm X_{ii}\leq P_i \\\forall i=1,2,\cdots,n\end{array} $X\in\mathbb{R}^{n\times n}$, $X>0$ ...
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0answers
49 views

Proof: A tangent space of the manifold of SPD matrices is the set of symmetric matrices

The set of SPD matrices, $\mathbb{P}_n := \{X \in \mathbb{R}^{n \times n} | X=X^T, X \succ 0 \} $, forms a differentiable manifold. Claim: The tangent space at a point, $A, T_A\mathcal{P}_n$ is the ...
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0answers
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Explain: “SPD matrices can be thought of as an extension of positive numbers”

So I am reading a paper (https://www.researchgate.net/publication/263699451_From_Manifold_to_Manifold_Geometry-Aware_Dimensionality_Reduction_for_SPD_Matrices) during which the author states that "SPD ...