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.
674 questions
2
votes
0answers
13 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}
$X\in\mathbb{R}^{n\times n}$, $X>0$ ...
1
vote
0answers
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 ...
1
vote
0answers
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 ...
0
votes
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
votes
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?
1
vote
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
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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?, ...
1
vote
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
$$
\begin{vmatrix}
A ...
0
votes
1answer
23 views
Why do these conditions ensure symmetric positive-definiteness?
Forgive me, if I have made a blunder or missed something obvious, I'm not a mathematician!
I'm trying to understand 2 seemingly simple lines of maths - and understand how the conclusions are drawn ...
0
votes
1answer
19 views
$\{x\mid x^TAx\leq 1\} = \{x\mid x^TBx\leq 1\} \Rightarrow A = B$, where $A\succ 0, B\succ 0$
I want show that $$\{x\mid x^TAx\leq 1\}_{\epsilon_A} = \{x\mid x^TBx\leq 1\}_{\epsilon_B} \Rightarrow A = B,$$ where $A\succ 0, B\succ 0$ (positive definite).
Can I prove this by arguing the ...
0
votes
2answers
18 views
Inside and outside of an ellipsoid
Let $A$ be a positive definite matrix.
The equation $x^TAx=1$ defines an ellipsoid.
I would like to justify the fact that $x^T A x > 1$ implies that $x$ is outside of the ellispoid.
In other ...
1
vote
0answers
54 views
Is the matrix $a_{ij} = \frac{1}{i+j-1}$ Positive Definite?
I have a matrix $A$ of size $n \times n$ defined as
$$a_{ij} = \frac{1}{i+j-1}$$
Where $a_{ij}$ is the $i^{th}$ row, $j^{th}$ column element of the matrix. So, $1 \leq i \leq n$ and $1 \leq j \leq n$....
1
vote
1answer
30 views
Positive Definiteness problem
Consider the positive definite matrix $B \succ 0$ and the matrix ( not necessarily square) $A$. what can we say abut the positive definiteness of:
$$ A^\prime B A$$
My hunch is that this is positive ...
1
vote
1answer
38 views
Does $0\prec B \prec A$ imply $A^{-1}B \prec I$
Does $0\prec B \prec A$ imply $A^{-1}B \prec I$?
I know:
$A,B$ are of the same size.
$A$, $B$ have strictly positive, real eigenvalues.
$A^{-1}B$ may not be symmetric since $A^{-1}$ and $B$ ...
1
vote
1answer
20 views
Ellipsoid representation by PSD matrix and by linear mapping
Consider the following two representations of ellipsoid:
$$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\}$$
If I want $E_1=E_2$, I ...
0
votes
0answers
30 views
Proof that Gauss- Seidel iteration method converges for any initial x if the matrix is self-adjoint and positive-definite
I am trying to create a direct proof that if the matrix A is self-adjoint and positive-definite, then the Gauss-Seidel iteration converges for any initial ${\bf x}_{0}$
I think I need to prove $\rho({...
6
votes
1answer
49 views
If a matrix $Q$ is symmetric and positie definite, is it possible to show that the matrix $Q-A^T(AQ^{-1}A^T)^{-1}A$ is also positive definite?
If I have a symmetric and positive definite $n\times n$ matrix $Q$ and a full row-rank totally unimodular $m\times n$, where $m<n$, matrix $A$, is it posible to show that the matrix
$$Q-A^T(AQ^{-...
3
votes
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 $x \in \mathbb{R^n}$....
1
vote
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 ...
0
votes
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 ...
0
votes
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
vote
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
votes
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
votes
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{...
0
votes
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 ...
0
votes
0answers
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?
2
votes
0answers
85 views
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}
\...
-1
votes
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.
1
vote
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 $A$ ...
0
votes
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, ...
0
votes
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 ...
-4
votes
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 ...
0
votes
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{...
0
votes
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/...
2
votes
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 ...
0
votes
0answers
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))
= ...
0
votes
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 $b^TCb>a^TCa$?
1
vote
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 $V$...
1
vote
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 ...
0
votes
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 ...
1
vote
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(\...
0
votes
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 ...
0
votes
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
$$\sigma(x)=|\psi(x)|^{...
5
votes
1answer
94 views
Trace inequality for a product of p.s.d. matrices and their pseudo inverse.
Let $A, B_i$ be positive semidefinite real matrices. Let $\dagger$ stand for the Moore-Penrose generalized inverse. I managed to prove that if
$\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
0
votes
2answers
18 views
If $A$ is positive definite and $Ax\le b$ can I say that $x\le A^{-1}b$?
If I have a vector enequiality $Ax\le b$ where $x,b\in\mathbb{R}^n$ and $A$ is a $n\times n$ positive definite matrix, can I say that the following vector inequality is true?
$$x\le A^{-1}b$$
0
votes
1answer
68 views
Hessian of negative log-likelihood of logistic regression is positive definite?
I'm trying to show that the Hessian of the negative of the log likelihood with two parameters is positive definite, but I'm not sure how to go about it once I compute the Hessian.
The function is:
$-...
0
votes
0answers
19 views
Show inequality for Euclidean norm on SPD matrix and identity matrix
Let A be a symmetric, positive definite matrix. Show that in the Euclidean norm $$||I-\frac{1}{\tau}A||_{2}<1$$ implies that $0<\tau<2||A||^{-1}_{2}$ for $\tau$ a scalar.
