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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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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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29 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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11 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 ...
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2answers
29 views

Proving Semidefinite From Adding Inverse and Subtracting Multiple of Identity Matrix.

Suppose A is an $n \times n$ positive definite matrix. Prove that $A+A^{-1}-2I_n$ is positive semidefinite. I know that the eigenvalues of $A^{-1} = \lambda^{-1}$, and that I have to relate that to ...
2
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2answers
26 views

Condition on $a$ for matrix to be positive definite

Assume you are given an $n\times n$ matrix with all elements equal to $a$, except for the diagonal values which are all $1$. What would be the condition on $a$ so that the matrix be positive definite?
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1answer
36 views

Sum of Symmetric Positive Definite Matrix and Scalar of Identity

If $A$ is an $n\times n$ symmetric positive definite matrix with the smallest eigenvalue $\lambda$, then for any $\mu>-\lambda$, $A+\mu I$ is positive definite. I am trying to show this, but I am ...
0
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1answer
25 views

Positive/Negative Definite/Semidefinite Test Generality

A test to determine whether a matrix is positive definite, negative definite, positive semidefinite, negative semidefinite, or none of the above, is to calculate the determinant of every cascading ...
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2answers
20 views

How to arrive at these conditions for 2x2 SPD matrices?

Claim: For the matrix $M = \left[ {\begin{array}{cc} a & c \\ c & b \\ \end{array} } \right]$ to be symmetric (trivial) and positive definite: $a>0$ and $ab-c^2>0$ where a,b ...
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2answers
48 views

Positive definite matrix implies the **infimum** of eigenvalues are positive (second version)?

I asked a similar question in here, but actually what I want to ask is more difficult as described below: Suppose $P(x): \mathbb{R} \to \mathbb{R}^{n \times n}$ is always a positive semi-definite ...
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1answer
18 views

Positive definite matrix implies the **infimum** of eigenvalues are positive?

Suppose $P(x): \mathbb{R} \to \mathbb{R}^{n \times n}$ is always a positive definite matrix, does it imply that the infimum (over $\mathbb{R}$) of the minimum eigenvalue of $P(x)$ is always positive?, ...
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1answer
26 views

Symmetric matrices property

Reading "Mathematical Physics: Classical Mechanics" by A. Knauf, I found the following statement: The positive symmetric matrices with determinant 1 can be written as $$ \begin{vmatrix} A ...
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1answer
23 views

Why do these conditions ensure symmetric positive-definiteness?

Forgive me, if I have made a blunder or missed something obvious, I'm not a mathematician! I'm trying to understand 2 seemingly simple lines of maths - and understand how the conclusions are drawn ...
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1answer
19 views

$\{x\mid x^TAx\leq 1\} = \{x\mid x^TBx\leq 1\} \Rightarrow A = B$, where $A\succ 0, B\succ 0$

I want show that $$\{x\mid x^TAx\leq 1\}_{\epsilon_A} = \{x\mid x^TBx\leq 1\}_{\epsilon_B} \Rightarrow A = B,$$ where $A\succ 0, B\succ 0$ (positive definite). Can I prove this by arguing the ...
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2answers
18 views

Inside and outside of an ellipsoid

Let $A$ be a positive definite matrix. The equation $x^TAx=1$ defines an ellipsoid. I would like to justify the fact that $x^T A x > 1$ implies that $x$ is outside of the ellispoid. In other ...
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0answers
54 views

Is the matrix $a_{ij} = \frac{1}{i+j-1}$ Positive Definite?

I have a matrix $A$ of size $n \times n$ defined as $$a_{ij} = \frac{1}{i+j-1}$$ Where $a_{ij}$ is the $i^{th}$ row, $j^{th}$ column element of the matrix. So, $1 \leq i \leq n$ and $1 \leq j \leq n$....
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1answer
30 views

Positive Definiteness problem

Consider the positive definite matrix $B \succ 0$ and the matrix ( not necessarily square) $A$. what can we say abut the positive definiteness of: $$ A^\prime B A$$ My hunch is that this is positive ...
1
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1answer
38 views

Does $0\prec B \prec A$ imply $A^{-1}B \prec I$

Does $0\prec B \prec A$ imply $A^{-1}B \prec I$? I know: $A,B$ are of the same size. $A$, $B$ have strictly positive, real eigenvalues. $A^{-1}B$ may not be symmetric since $A^{-1}$ and $B$ ...
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1answer
20 views

Ellipsoid representation by PSD matrix and by linear mapping

Consider the following two representations of ellipsoid: $$E_1 = \{x \mid x^TSx\leq 1, \, S \succ 0\}$$ and $$E_2 = \{y \mid y = Ax, \, \|x\|\leq 1, \, \det(A) \neq 0\}$$ If I want $E_1=E_2$, I ...
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30 views

Proof that Gauss- Seidel iteration method converges for any initial x if the matrix is self-adjoint and positive-definite

I am trying to create a direct proof that if the matrix A is self-adjoint and positive-definite, then the Gauss-Seidel iteration converges for any initial ${\bf x}_{0}$ I think I need to prove $\rho({...
6
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1answer
49 views

If a matrix $Q$ is symmetric and positie definite, is it possible to show that the matrix $Q-A^T(AQ^{-1}A^T)^{-1}A$ is also positive definite?

If I have a symmetric and positive definite $n\times n$ matrix $Q$ and a full row-rank totally unimodular $m\times n$, where $m<n$, matrix $A$, is it posible to show that the matrix $$Q-A^T(AQ^{-...
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2answers
58 views

Alternative definition of positive definite matrix [duplicate]

In a text book I am following, there is a definition of positive definite matrices, which I did not see before: $x^TAx \geq \alpha x^Tx$ where $\alpha$ is a positive scalar and $x \in \mathbb{R^n}$....
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1answer
20 views

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$

If real Matrix A is symmetric and positive definite then $X^TAY $ represent dot product with respect to basis of $\mathbb R^n$ I am studying now bilinear form .I wanted to prove above theorem. I ...
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0answers
18 views

How to find rectangular matrix $A$ such that $A*B $is positive definite for a given rectangular matrix $B$?

Given B, A rectangular $m \times n$ matrix. I would want to find a rectangular matrix; $A_{m \times n }$ such that $AB$ is positive/negative definite. there exists No further assumptions on the ...
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0answers
27 views

A problem on positive definite matrix

I am studying a chapter on positive definites and there is this question in which I have to find whether the quadratic forms are positive definite or not. I have to confirm my answer for this ...
1
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1answer
30 views

Prove this matrix inequality

I am reading a math paper where a proof uses the following result without proof. I have been stuck on this part for various days, and have talked to others about it but still can't figure out why it ...
0
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1answer
18 views

For what kind of matrix $A$, there is a (symmetric) positive definite matrix $B$ such that $BA$ is symmetric

Let $A$ be a $n\times n$ real matrix. My ultimate goal is to find a sufficient condition on $A$ such that all the eigenvalues of $A$ are real. Therefore, I want $A$ to be self-adjoint with respect ...
0
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1answer
19 views

Can a positive-definite diagonal matrix have square roots that are positive-definite but not diagonal?

Consider a a diagonal $n\times n$ matrix $M$ with real diagonal elements $m_{11},m_{22},\dots,m_{nn}>0$. I am interested in finding positive-definite $n \times n$ matrices $X$ that solve \begin{...
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1answer
25 views

Symmetric and definite positive matrix - complex vs real coefficients

Let $M$ be a symmetric square matrix with complex coefficients, such that its imaginary part $N$ is positive definite. Is it true that $$A := {^t M} N^{-1} \overline M $$ has real coefficients? (Here ...
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22 views

Known conditions to make $A \otimes B$ be pos.def., even if $A$ is not pos.def.?

Are there conditions that I should demand to be sure that $A \otimes B$ is positive definite, even when allowing $A$ not being positive definite, while $B$ is positive definite?
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What is the maximum value of $x^TAx$ subject to $x\in\{\pm1\}^n$?

Let $A \in \mathbb{R}^{n\times n}$ be symmetric and positive definite. What is the following maximum? $$\max_{x\in\{\pm1\}^n}x^T A x$$ My attempt: if all $a_{ij}\geq 0$, then \begin{equation} \...
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1answer
42 views

Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable. [closed]

Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Then $AB$ is diagonalizable. I want to see if this is true or false.
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2answers
40 views

Conditions for a certain parametric matrix to be positive definite

The task: Let's consider matrix $A=\begin{bmatrix}1&a&a\\a&1&a\\a&a&1 \end{bmatrix}$. Show that $A>0$ if and only if $-1<2a<2$. Solution attempt: By definition $A$ ...
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1answer
32 views

bilinear form only positive or negative

Let $f$ be a definite, symmetric bilinear form. Show that $f$ is positive or negative. Consider the quadratic form $q(x,y,z)=x^2+y^2-z^2$, then the polar form assocaited to $q$ is symmetric, ...
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1answer
14 views

For a positive semi-definite $d\times d$ matrix $A$, $ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $ for every $S\in\text{SPD}_{d}$.

For a positive semi-definite $d\times d$ matrix $A$, $$ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $$ for every $S\in\text{SPD}_{d}$. I would like to show the above statement. If ...
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2answers
162 views

Prove the following set of unit vectors is orthogonal.

Suppose ${v_1 , v_2 , ……..v_n}$ are unit vectors in $\mathbb R^n$ such that $ || v||^2 = \sum_{n=1}^{\infty} | <v_i , v>|^2 $ for all $v \in\mathbb R^n $ Then I have to prove that ...
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3answers
43 views

Show that the matrix is positive definite

We have the tridiagonal matrix $A=\begin{pmatrix}2 & 1 & \ldots & 0 \\ 1 & 2 & 1 & \ldots \\ \ldots & \ldots & \ldots & \ldots \\ 0 & \ldots & 1 & 2\end{...
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0answers
14 views

Is LDLT decomposition of stochastic matrices stable?

Matrix A has: non-negative entries columns that each sum to 1. LDLT decomposition requires the matrix A to be positive or negative semi-definite to be stable. https://eigen.tuxfamily.org/dox/...
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1answer
50 views

The true meaning of eigenvalues and eigenvectors, and positive (semi) definite matrix?

After studying some linear algebra, the true meaning of eigenvalues, eigenvectors, and positive definite matrix are still ambiguous. So, could you answer my questions or explanations about those ...
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35 views

Ways to check positive definiteness of bilinear form

For each real number $\alpha$, we define the bilinear form $F_{\alpha}:\mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R} $ by $\displaystyle F_{\alpha}(((x_1, x_2, x_3), (y_1, y_2, y_3)) = ...
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0answers
6 views

How to prove monotonicity of symmetric positive definite quadratic form

Let $a=(a_1,...,a_n) \in \mathbb{R}^n$, $a_1>0$ and $h>0$. Define $b=(a_1+h,...,a_n)$. Furthermore let $C$ be a positive definite symmetrix matrix How can i proof that $b^TCb>a^TCa$?
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1answer
37 views

Positive Definite Matrix Norm

I am attempting to prove that a norm where for any $(k \times 1)$ vector x with $(1 \times k)$ transpose y: $$ \|x\|=\sqrt{yVx} $$ for some $(k \times k)$ positive definite (and symmetric) matrix $V$...
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0answers
36 views

Question about the inverse of a positive definite matrix? [closed]

So I was reading the book on Graphical Models by Joe Whittaker. And, in chapter 5 of the book, there is a question about showing that the diagonal elements of the inverse of a positive definite matrix ...
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2answers
34 views

LU decomposition of SPD matrix without partial pivoting?

I get why diagonal dominant matrices do not need partial pivoting before Gaussian elimination can be applied in order to gain a LU decomposition, but why is this also the case for SPD matrices in ...
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1answer
37 views

integral involves positive definite function and Bessel function

Could the following integral be $0$? \begin{eqnarray*} \lim_{x \to 0^+} \int_0^\pi \vartheta^{\alpha+\frac{1}{2}} \frac{J_{\alpha-\frac{1}{2}} ( x \vartheta )}{x^{\alpha-\frac{1}{2}}} g(\...
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0answers
11 views

Minimize double dot product subject to positive definite constraints

Given a symmetric matrix $\varepsilon$ and a 4th order tensor $C$, how can we find the matrix $\delta$ that minimizes $C{:}(\varepsilon{+}\delta){:}(\varepsilon{+}\delta)$ subject to the constraints ...
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0answers
15 views

For some positive definite function $\sigma(x)$ does there always exist $\psi(x)$ such that $\sigma(x)=|\psi(x)|^{2}$???

Ok, this seems like common sense to me but I just had to check. Given some positive definite function $\sigma(x)$ can we always find $\psi(x)$ such that we can represent it as $$\sigma(x)=|\psi(x)|^{...
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1answer
94 views

Trace inequality for a product of p.s.d. matrices and their pseudo inverse.

Let $A, B_i$ be positive semidefinite real matrices. Let $\dagger$ stand for the Moore-Penrose generalized inverse. I managed to prove that if $\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
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2answers
18 views

If $A$ is positive definite and $Ax\le b$ can I say that $x\le A^{-1}b$?

If I have a vector enequiality $Ax\le b$ where $x,b\in\mathbb{R}^n$ and $A$ is a $n\times n$ positive definite matrix, can I say that the following vector inequality is true? $$x\le A^{-1}b$$
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1answer
68 views

Hessian of negative log-likelihood of logistic regression is positive definite?

I'm trying to show that the Hessian of the negative of the log likelihood with two parameters is positive definite, but I'm not sure how to go about it once I compute the Hessian. The function is: $-...
0
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0answers
19 views

Show inequality for Euclidean norm on SPD matrix and identity matrix

Let A be a symmetric, positive definite matrix. Show that in the Euclidean norm $$||I-\frac{1}{\tau}A||_{2}<1$$ implies that $0<\tau<2||A||^{-1}_{2}$ for $\tau$ a scalar.