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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How to estimate the spectralnorm by the infinity norm?

Let $A \in \mathbb{R}^{n \times n}$ be a symmetric, positive-definite matrix. Furthermore let $A$ be diagonally dominant, i.e. $\max_{i=1,...,n} \sum_{j=1, j\neq i}^{n} \frac{|a_{ij}|}{|a_{ii}|} \leq ...
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Lower bound on the off-diagonal elements of a PSD matrix

Suppose we have a PSD matrix $X\in\mathbb{R}^{2d}$, which could be written in the following block form $$X=[X_1\quad X_2;\quad X_2^\top\quad X_3],$$ where $X_1, X_3\in\mathbb{R}^d$ are PSD matrices, ...
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Let $B$ be simetric and positive definite. Show that $x^T\left(B - \frac{Bss^TB}{s^TBs}\right)x $ is also positive definite

Let $B\in \mathbb{R}^{n\times n}$ simetric and positive definite and $s\in\mathbb{R}^n-\{0\}$ Let $$M := B - \frac{Bss^TB}{s^TBs}$$ Show that $$x^TMx > 0$$ My try: $$x^TMx$$ This is $$x^T\left(B - \...
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The miraculous nature of the "matrix coefficients" $\langle Tv,v \rangle$ (especially in the context of positive type functions)

$\newcommand{\ak}[1]{\langle #1 \rangle}$I've noticed that for linear operators $T$ and an inner product $ \ak{\bullet, \bullet }$, the expression $\ak{ Tv,v}$ tends to show up a lot. For instance, it ...
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When does Inverse Fourier transform look close to a positive definite function?

Let $G$ be a commutative locally compact group, and $\hat{G}$ be its dual group, consisting of all continuous characters (continuous homomorphisms from $G$ to the circle group $\mathbb{T}$) . I can ...
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6 votes
3 answers
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Criteria for $3 \times 3$ matrix to positive definite

Here it is said that a $2\times 2$ matrix $A$ is positive definite if and only if $tr(A) >0$ and $det(A)>0$. This will not work if $A$ is $3\times 3$. But is there any way to enforce the ...
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Prove that there is a positive real number $\lambda$, so that $A =\lambda B$, for two positive definite square matrices [closed]

A and B are positive definite square $n$ $\times$ $n$ matrices. The thereby defined dot products define the same orthogonality relation. Which means: $∀v, w ∈ R^n : v · Aw = 0 ⇔ v · Bw = 0.$ Show that ...
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When is a sum of products of two matrices and the transposes positive definite?

Let $X$ and $Y$ be $n \times m$ matrices. The matrix $$ A = X^TY + Y^TX $$ will be a $m \times m$ square, symmetric matrix. Is it possible to say: i) when is $A$ is positive definite? and when it is ...
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How to check whether this matrix is positive semidefinite?

Given the following symbolic matrix \begin{equation*} A= \begin{pmatrix} -\cos(\theta_1-\theta_2)&\cos(\theta_1-\theta_2)&0 \\ \cos(\theta_1-\theta_2) & -\cos(\theta_1-\theta_2)-\cos(\...
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Change of eigenvalues under near orthogonal matrix multiplication

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, ...
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If Hermitian $A$ is positive definite, is this block matrix also positive definite?

Let $A$ be an Hermitian matrix, and suppose $A$ is positive definite, i.e., $(Ax, x)>0$ for all $x\in \mathbb C^n$. If I let $A=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{...
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On the positive definiteness of the observability Gramian

Given the system $\dot{x} = Ax$, $y = Cx$, it is known that the Gramian, given by $$ W({t_0},{t_1}) = \int_{t_0}^{t_1} e^{A^T(\tau -t_0)}C^TC e^{A(\tau -t_0)} \,{\rm d}\tau $$ is positive definite for ...
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Determine whether smooth symmetric bilinear form is definite over the trivial normal bundle of the oblique manifold

Given a smooth symmetric bilinear form over the trivial normal bundle of the oblique manifold $\mathcal{OB}(n,m)$ imbedded in $\mathbb{R}^{n \times m}$ is there an efficient way to determine whether ...
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How I can convert a negative definite matrix to a positive definite?

If I have a negative definite matrix $A$, can I convert this matrix to a positive definite by taking columns $A$ of the negative eigenvalues?
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Spectral properties of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)}~~~~~~~~~~~~~~~...
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Matrix norm ordering upon multiplication by positive definite matrix

Assume I know the following to be true \begin{equation} ||A||>||B|| \end{equation} where $A$ and $B$ are $n\times n$ matrices and the norm can be any norm. If I define a positive definite matrix $...
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Averaged matrix exponential inverse

Let $H$ be a given symmetric matrix, i.e., $H\in\mathbb{R}^{d\times d}_{\rm sym}$, $d\geq1$. The operator $T$ is given by $$TX=\int_0^1e^{(1-s)H}Xe^{sH}\mathrm{d}s,\qquad X\in\mathbb{R}^{d\times d}_{\...
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Conditions for a block matrix with parameters to be positive definite

Let $A$ and $B$ be $n \times n$ positive definite matrices. Let $C$ be any $m \times n$ matrix. Can we always find $\alpha$ and $\beta$ such that the matrix $$\begin{bmatrix} A + \alpha C^\top C + \...
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insert a matrix in-between product of vectors

Let $a=[1,2]^t$ ( $^t$ means transpose) and let $A=\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$. One can verify that $a^t a = 5$, $a^t A a = 14$, and $a^t a \cdot \det(A)= 15$. So obviously, $...
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How to use barrier method for constraints like $X \succ 0$?

When reading about interior-point methods in Stephen Boyd & Lieven Vandenberghe's Convex Optimization, a question arose about how to use barrier method for the constraint $X$ is positive definite, ...
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How to carry out the expected value of the cost function in a LQG problem to tackle path tracking?

I have a system, whose state is defined by $x_t$. The transition state mapping (STP) for the systems is defined as: $$x_{t+1} = A_t x_t + B_t u_t + w_t$$ where, $x \in \mathbb{R}^{n \times 1}$, $A_t \...
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Is there any hint how to prove this?

Let's consider matrix M is defined as follows: $M = \begin{bmatrix} P & v \\ v^t & d \end{bmatrix}$, where $P \succ 0$, d is a scalar, and v is a vector. Problem: :To $M \succ 0$ be a ...
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1 vote
1 answer
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Question about positive definite matrix and inequality proof

Problem: Let $X$ and $R$ be positive definite matrices, $C$ is a matrix of compatible dimension, and define $g(X)$ as $g(X)=X-XC'[CXC'+R]^{-1}CX$ Prove that if $X>Y>0$, then $g(X)>g(Y)$. From ...
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How to show that all of the all eigen values of the matrix are positive?

I have a matrix \begin{bmatrix} 6I &-4I&0&0&......0\\ -4I &6I& -4I & 0&......0\\ 0 & -4I & 6I & -4I &......0\\ .\\ .\\ .\\ 0&0&0&........-4I&...
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What functions preserve symmetry and positive-definiteness of covariance matrices?

Suppose I have covariance matrices $H_1,…,H_n$ (symmetrical, positive-definitive) and corresponding weights $w_1,..,w_n$. We want to find a function $f$, such that $H = f(H_1,…,H_n,w_1,…,w_n)$ is ...
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Confidence interval for matrix inverse [closed]

Let $\mathbb S_{++}$ denote the set of symmetric positive definite matrices. Suppose that $X \in \mathbb S_{++}$ and that I have a confidence interval for $\hat X \in \mathbb S_{++}$, i.e., $$ \mbox{...
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What is meant by a matrix being strictly positive definite on its range?

Consider the following matrix $$A= \begin{pmatrix} a_{11}&0&0&0&0&0\\ 0&a_{22}&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&a_{44}&0&0\\...
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row norms of positive definite matrices

Suppose $A$ is symmetric positive definite. Is there any relationship between the row norm (excluding the diagonal) $$ \left( \sum_{j\neq i} |A_{i,j}|^p \right)^{1/p} $$ and the diagonal entry $A_{i,i}...
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2 votes
1 answer
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Conditions for positive definite Symmetric Toeplitz matrix with powers

Let $A_n$ be the following matrix for $p\leq 1$: \begin{equation} A_n = \begin{bmatrix} 1 & 2^{-p} & 3^{-p} & \dots & n^{-p} \\ 2^{-p} & 1 & 2^{-p} & \dots & (n-1)^{-p} ...
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Is it possible that $A+A_{\text{off-diag}}\succ0$ , where $A\succneqq0$?

Denote $A_{\text{off-diag}}:=A-\text{diag}(A)$, i.e. setting diagonal elements to be zero. Denote $A\succneqq0$ iff $A$ is positive semidefinite (and not positive definite), while $A\succ0$ iff $A$ is ...
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Invertiblity of sum of matrices

Assume two matrices A and B are both symmetric positive definite. I was wondering whether I+AB and I-AB are invertible, where I is the identity matrix. (A hint might be AB is positive definite iff AB ...
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5 votes
1 answer
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If $\lambda(\mathbf{A}+\mathbf{B})=\lambda(\mathbf{A})+\lambda(\mathbf{B})$, are HPD matrices $\mathbf{A}, \mathbf{B}$ simultaneously diagonalizable?

$\mathbf{A}, \mathbf{B} \in \mathbb{C}^{n \times n}$ are Hermitian and positive-definite (HPD) matrices. The following conditions are equivalent: $\mathbf{A}$ and $\mathbf{B}$ commute. $\mathbf{A}$ ...
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1 answer
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There exists a positive definite $Av=w$ if and only if $v$ and $w$ are not orthogonal

Let $v,w \in \mathbb R^n$ be two vectors with $v^Tw \neq 0$. Prove that there exists a positive-definite matrix $A$ such that $Av=w$. The converse is trivial, but how do I construct a positive-...
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What is the difference between regularization and preconditioning for a matrix?

The literature says "Preconditioning is a technique for improving the condition number of a matrix. Suppose that $M$ is a symmetric, positive-definite matrix that approximates $A$, but is easier ...
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Is the product P = B^T * B positive-definite if B is a square matrix?

I am looking for a matrix parametrization that guarantees positive definiteness. The target matrix P shall be positive-definite. I thought about writing P = B^T * B, where B is some square matrix. ...
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1 answer
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Positive definite matrix with positive entries

I have a question about linear algebra. My question is: Is any real symmetric matrix $A$ with non-negative entries such that $\det(A)>0$ positive definite? Thank you.
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Semipositive definite and hermiticity implication

Its something that is in all basic algebra books that, a semipositive definite operator is Hermitian and the eigenvalues of this hermitian operator are positive. But I couldn't find any place where ...
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1 answer
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Discrete spectrum for product of operators

Let $A$ and $B$ be two positive-definite self-ajoint operators on a Hilbert space $H$. If the spectrum of both $A$ and $B$ is discrete, can we affirm that the spectrum of $AB$ will be also discrete ? ...
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0 answers
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Positive semidefiniteness of matrix $\mathbf{G}$

Suppose $\mathbf{X}$ and $\mathbf{Y}$ are square, real, and symmetric matrices. $\mathbf{X}$ is positive definite and $\mathbf{Y}$ is positive semidefinite. $a$ and $b$ are positive scalars, and $\...
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1 vote
1 answer
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Solving for a Diagonal Positive Definite Matrix: $\Delta$ such that $b'\Delta C'(C\Delta C')^{-1}=a'DC'(CDC')^{-1}$

Any help with the following conjecture would be highly appreciated. I have a hard time figuring out how to start. Though counterexamples would be certainly helpful, if the conjecture is not always ...
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1 vote
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Decomposition of positive definite matrix.

Let $A$ be a positive definite matrix. Following are the decomposition's of matrix $A$: $A = PP$ $A = MM^{T}$ Empirically we found that $||P||_{F}^2 = ||M||_{F}^2$ but how to prove this ...
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Inverse of a positive symmetric matrix with large entries

I have a random vector $Z = X_{1} + X_{2} + \ldots + X_{T}$, where each $X_{i}$ is a $n\times 1$ random vector with finite mean and covariance matrix. Each $X_i$ has a different mean and covariance ...
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Deriving a Cholesky decomposition algorithm

I am studying numerical linear algebra using Lloyd Trefethen's book. In the chapter on systems of linear equations, I am reading about the Cholesky decomposition. I understand existence and uniqueness,...
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Uniform sampling on the set of symmetric positive-semidefinite matrices with bounded entries

What is the most correct way to randomly generate a (square) symmetric positive-definite matrix $A$ with nonnegative entries bounded in [0,1]? One way I can think of is by sampling a matrix $X$ from ...
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1 answer
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Proof of "If M and N are positive definite, then the products MNM and NMN are also positive definite"

Here on Wikipedia, it states that: "If M and N are positive definite, then the products MNM and NMN are also positive definite" I've tried looking for a proof of this statement, but I cannot ...
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How does one denote the set of all positive-definite square matrices? [duplicate]

For example, can I write: The matrix $X \in \mathbb{R}^{p \times p}_{>0}$ follows a Wishart distribution $$ X \sim \mathcal{W}(V,n) $$ where $\mathbb{R}^{p \times p}_{>0}$ is the set of all ...
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0 answers
33 views

Under what conditions can we exchange the order of a matrix multiplication?

I am wondering what are the necessary conditions for matrix multiplication to be commutative, especially those related to positive (semi) definite matrices.
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0 answers
40 views

Why eigenvectors of a symmetric toeplitz matrix has a particular property?

I got a positive-definite symmetric toeplitz matrix $\mathbf{A}$, for example, $\begin{equation} \mathbf{A} = \left[ \begin{array}{ccc} 1 &2 &3 \\ 2& 1& 2 \\ 3& 2& 1 \end{...
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3 votes
1 answer
69 views

Sum of matrix inverses

I am trying to compute the following finite sum: $$\sum_{i=1}^n\left(A+\lambda_iB\right)^{-1}$$ In this sum, $A$ and $B$ are are positive-definite matrices (so they are inversible). The $\lambda_i$ ...
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3 votes
1 answer
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Trace inequality on positive definite matrices

My question is the following: Suppose that $A, B$ are symmetric positive definite real matrices. Is it true that $$ \mathrm{trace}\big((A+B)^{-1}\big)\geq \frac{1}{2} \min\Big\{\mathrm{trace}(A^{-1}),...
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