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10:00 AM
A new tag was created by Shootforthemoon.
Q: Example showing the implicit function theorem is not a double implication?

ShootforthemoonIs there example of let's say a two variable function that can be rewritten as an explicit function of one variable but that does not satisfy the assumptions of the implicit function theorem? In fact, if the theorem only provides sufficient conditions, we should say that there exists some implic...

In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. == Formal statement == If X is a compact topological space, and { fn } is a monotonically increasing sequence (meaning fn(x) ≤ fn+1(x) for all n and x) of continuous real-valued functions on X which converges pointwise to a continuous function f, then the convergence is uniform. The same conclusion holds if { fn } is monotonically decreasing instead of increasing. The...
Two new tags and were cretaed by Ongky Denny Wijaya.
Q: Finding mixed probability density function. Please check my answer.

Ongky Denny WijayaGiven random variable $X$ with cumulative distributive function $$ F_X(x)= \begin{cases} 0&x<0 \\ \dfrac{1}{4}x^2&0\leq x<1 \\ \dfrac{1}{2}&1\leq x<2 \\ \dfrac{1}{3}x&2\leq x<3 \\ 1&x\geq 3 \end{cases}. $$ Find the probability density function of $X$. To find the p.d.f, I plot the graph of $F_X...

A new tag was created by miosaki.
Q: is it a metric on the probability measure

miosaki\begin{align} D(m_1,m_2)=\sup\limits_{G\in \mathcal L_{p}}\left[\int\limits_{\mathbb R} G dm_1-\int\limits_{\mathbb R} G dm_2\right] \end{align} where for $p\in (0,1]$ and \begin{align} \mathcal L_p=\{f : \mathbb R\to \mathbb R: |G(x)-G(y)|\le D^{p}(x,y)\}. \end{align} Clearly (1) $D(m, m)=0$ f...

Q: A triangle Inequality

YNKIf $\pmb a$, $\pmb b$, and $\pmb c$ are the three sidelengths of an arbitrary triangle, prove that the following inequality is true with the equal sign holding for equilateral triangles. Inequality in Compact Form: $$ 7\left(a+b+c\right)^3-9\left(a+b+c\right)\left(a^2+b^2+c^2\right)-108abc\ge0$$...

It seems that triangle inequality briefly existed in 2013: math.stackexchange.com/posts/432938/revisions
Queries which show also editors who added/removed the tag: data.stackexchange.com/math/query/1105163/… data.stackexchange.com/math/query/1038474/…
A new tag was created by Na'omi.
Q: Feasible directions on the FDA-NCP

Na'omiI am studying the Feasible Directions Algorithm to Nonlinear Complementarity Problem FDA-NCP (see http://ns.optimize.ufrj.br/files/SandroRodriguesMazorche.pdf). At page 31, we have: $(F_i(x^k)e_i+x_i^k\nabla F_i(x^k))d^k_1=0$. where $x^k,e_i,F$ are $n$-dimensional vectors and $x_i,F_i$ the...

A new tag was created by Rodrigo de Azevedo. He also created a tag-excerpt. There are now 13 questions with the tag.
Q: If $A$ is a Hermitian matrix then $SAS^*$ is also Hermitian

Penelope Clearwater If $A$ is an $n \times n$ Hermitian matrix, and $S$ is an $n \times n$ matrix, then $S A S^*$ is also Hermitian. Why is this true? I have seen this claim made in several places but can't find a proof.

Q: Prove sum of products of Hermitian matrices to be Hermitian

SimonA given problem states: If $A$ and $B$ are Hermitian matrices, prove that $(AB+BA)$ is Hermitian. Because the sum of Hermitian matrices is known to be Hermitian, the problem seems to me to boil down to proving that the products $AB$ and $BA$ are Hermitian. But this isn't necessarily so, unl...

Q: Condition that multiplied Hermitian matrix stays Hermitian

madison54Suppose we are given a Hermitian matrix $E \in \mathbb{C}^{n\times n}$. I want to find sufficient conditions on the entries of a real symmetric matrix $M$ (depending on the entries of the given matrix $E$) such that the matrix $EM$ is again Hermitian. Is there a systematic way to determine the ...

Q: Relationship between hermitian matix and hermitian transformation

KeinMy TA said that every hermitian matix implies transformation is hermitian because you can find orthonormal basis for every hermitian matrix and therefore transformation is hermitian. Is that true?? What about the converse? If the transformation is hermitian, then matrix of T is hermitian?

Q: Prove the following result for Hermitian and Skew-Hermitian matrix

MKS If $H$ be a Hermitian matrix, prove that $\det H$ is real number. If $S$ be a skew Hermitian matrix of order $n$, prove that (i). if $n$ be even, then $\det S$ is real number; (ii). if $n$ be odd, then $\det S$ is a purely imaginary number or zero. Attempt: 1. Let $H=P+iQ$ be a Hermitian ...

Q: Orthonormal basis for Hermitian matrix

DanielSuppose there is a hermitian matrix. Then, Can we always find out orthonormal basis for this matrix ? And, Is there any relationship between hermitian matrix and hermitian transformation? If matrix is hermitian, does that imply transformation is hermitian? or If the transformation is her...

Q: Generating random commuting hermitian matrices

TarekHow can I generate random commuting hermitian matrices ? EDIT: Another question: given a certain hermitian matrix, how can I generate a random hermitian matrix which commutes with it?

Q: Matrices which are both unitary and Hermitian

M.S. DoustiMatrices such as $$ \begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} \cos\theta & i\sin\theta \\ -i\sin\theta & -\cos\theta \end{bmatrix} \text{ or } \begin{bmatrix} \pm 1 & 0 \\ 0 & \pm 1 \end{bmatrix} $$...

Q: Eigenvectors of a Hermitian matrix

Jakub WagnerA Hermitian matrix is a complex square matrix which is equal to its conjugate transpose. Its matrix elements fulfil following condition: $$a_{ij} = \bar{a}_{ji}$$ Everytime, I compute eigenvectors of a Hermitian matrix using Python, the first coefficient of the eigenvector is a pure real number...

Q: Hermitian matrix has positive eigenvalues

user247618I understand that Hermitian matrices has real eigenvalues. Just to hit the point home, I have the following question. Does every Hermitian matrix has eigenvalues? Since the proof assumes that the eigenvalue exists, the proof does not imply that every Hermitian matrix must have some eigenvalues....

Q: Product of two Hermitian matrices

titoEarlyAccording to Wikipedia: The product of two Hermitian matrices $A$ and $B$ is Hermitian if and only if $AB = BA$. So if I understood correctly, if $C=AB$, then C will be Hermitian if and only if $AB=BA$. But... I've been able to create a matrix $S$ then did $R=SS^H$, and $R$ turned out to...

Q: If A is unitary and $\det (A^H) = \det(conjugate(A))$ and $\det(A) \det(conjugate(A)) = \det(A)^2$ Why can't I say that $det(A^H) = +/- det(A)$

oliverthis came up on a Homework I have. I had to prove that the absolute value of the determinant of a Unitary Matrix is 1. So because $\det (A^H) = \det(conjugate(A))$ and $\det(A) \det(conjugate(A)) = \det(A)^2$ as well as that for a Unitary Matrix $A*A^H=I$ that $\det(A^H) = \pm \det(A)$ This w...

3 hours later…
1:36 PM
A new tag was created by Rodrigo de Azevedo. He also created a short tag-excerpt.
Q: Perron–Frobenius theorem

user153012What exactly is the Perron–Frobenius theorem? In different books and papers I read different statements, and I don't know what the truth is. In Wikipedia there are also a lot of statements under this label. And if somebody can characterize the statement for me, then I need also a proof. I read t...

Q: Prove that the eigenvalue with largest absolute value is positive and real

Quốc Nguyễn Minh Let $A \in \mathbb R_{>0}^{n \times n}$. Consider all the eigenvalues of $A$, including complex-valued ones. Prove that the eigenvalue that has the largest absolute value is a positive real eigenvalue. I also expect the proof using just elementary linear algebra.

5 hours later…
6:18 PM
A new tag was created by Nathaniel.
Q: Is an exponential family the same as an exponentially convex set?

NathanielAn exponential family of probability distributions is usually defined as $$ p(x) = e^{\theta\cdot f(x) \,-\, \psi(\theta)}, $$ where $\theta$ is a vector of parameters, $f(x)$ is some arbitrary vector-valued function of $x$, and $\psi= \log\sum_x e^{\theta\cdot f(x)}$, which ensures normalisation...

A new tag was created by heepo. The same user also created a tag-excerpt.
Q: A Third Derivative in Ahlfors' `Some Remarks on Teichmuller Space'

P7EI'm slugging my way through Ahlfors' "Some Remarks on Teichmuller Space" and am stuck on the calculation of equation (1.18). The problem here is to compute the third derivative, on the unit disk $\mathbb{D}$, of the function $$ \Phi(\zeta) = \frac{-2}{\pi} \iint_{\mathbb{D}} \bar{\nu}(z) \frac{\z...

However, there is an already existing tag . Should there be a synonym, or should the newly created tag be simply removed?

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