0
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...
1
Given 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...
4
2
If $\pmb a$, $\pmb b$, 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:
$$ 7\left(a+b+c\right)^3-9\left(a+b+c\right)\left(a^2+b^2+c^2\right)-108abc\ge0$$...
0
2
0
1
0
2
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 ...
0
1
32
3
A 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...
0
I 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....
4
this 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
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.
2
An 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...
1
