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.
1
vote
1answer
37 views
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 ...
0
votes
1answer
17 views
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 ...
1
vote
0answers
29 views
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) \\ ...
0
votes
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?
...
1
vote
3answers
38 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
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
$$
\lambda_{\mathrm{min}} ...
1
vote
1answer
30 views
How to prove that $\langle P,A^2 \rangle \le 0 $ for every positive $P$ and skew-symmetric $A$?
I have stumbled upon the following claim, and I wonder if it has a simple proof:
Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\...
0
votes
0answers
9 views
Approximation of a positive definite matrix
I have a covariance matrix (A), which is positive definite. I would like to approximate matrix A by another positive definite matrix B in such a way, that the eigenvalues of B span only 2 orders of ...
1
vote
3answers
43 views
Principal Minor Theorem for dimension 2
Let $A$ be a real ($2 \times 2$)-matrix such that the map $\mathbb{R}^2 \times \mathbb{R}^2 \to \mathbb{R}$, $\langle x,y \rangle = x^t A y$ is a scalar product. Now I would like to show that the ...
0
votes
1answer
45 views
how to prove the following linear matrix inequality
$K=\{1,2,\dots,m\}$
$A_1,\dots, A_m$ be any square matrices. $n\times n$ say.
If there exist positive definite symmetric matrices $P_1,\dots, P_m$, $(n\times n)$ say.
such that, $A_i^TP_jA_i-P_i<...
1
vote
0answers
36 views
Compare ratio of two determinants before and after adding sum of rank 1 matrices in both determinants
I have a ratio between two determinants:
$$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 $...
2
votes
2answers
63 views
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 ...
0
votes
1answer
19 views
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. ...
1
vote
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?
0
votes
0answers
16 views
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$ ...
1
vote
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)$ ...
1
vote
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 ...
1
vote
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,...
3
votes
2answers
54 views
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}$, $...
0
votes
0answers
12 views
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 ...
6
votes
1answer
48 views
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 ...
0
votes
0answers
20 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 ...
0
votes
0answers
10 views
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)$ ...
1
vote
2answers
27 views
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 ...
1
vote
0answers
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 ...
3
votes
2answers
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 \...
0
votes
2answers
39 views
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 ...
3
votes
2answers
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 ...
0
votes
1answer
35 views
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 ...
0
votes
0answers
23 views
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 ...
1
vote
1answer
23 views
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 ...
0
votes
0answers
20 views
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{...
0
votes
3answers
57 views
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 ...
1
vote
0answers
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\...
1
vote
1answer
29 views
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 ...
1
vote
1answer
29 views
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}$...
0
votes
0answers
17 views
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 ...
2
votes
2answers
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}
...
2
votes
0answers
45 views
Is there a closed-form formula for the derivative of the positive factor of a matrix?
$\newcommand{\psym}{\text{Psym}_n}$
$\newcommand{\sym}{\text{sym}}$
$\newcommand{\Sym}{\operatorname{Sym}}$
$\newcommand{\Skew}{\operatorname{Skew}}$
$\renewcommand{\skew}{\operatorname{skew}}$
$\...
1
vote
1answer
44 views
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 ...
0
votes
0answers
13 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?
0
votes
0answers
44 views
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, &...
1
vote
2answers
47 views
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 ...
2
votes
2answers
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 ...
0
votes
0answers
25 views
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 & \...
2
votes
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 ...
0
votes
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?
2
votes
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}
$X\in\mathbb{R}^{n\times n}$, $X>0$ ...
2
votes
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 ...
2
votes
0answers
12 views
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 ...
