Tagged Questions
1
vote
1answer
51 views
How do I show that these sums are the same?
My textbook says that I should check that
$$ \sum_{i=0}^\infty \frac{\left( \lambda\mathtt{I} + \mathtt{J}_k \right)^i}{i!} $$
is in fact the same as the product of sums
$$ \left( \sum_{i=0}^\infty ...
3
votes
2answers
97 views
Linear Algebra Matrix Question
I am having trouble showing that $e^AX = Xe^A$ for all $n$ by $n$ matrices $X$ where $A$ is an invertible $n$ by $n$ matrix iff $AX = XA$ for all $X$.
Any help will be appreciated. Thank you
1
vote
0answers
51 views
Power Series and Matrices
I am trying to prove that if a function $f(x)$ can be written as a power series in the form
\begin{equation}
f(x)=\sum_{n=0}^{\infty}c_n(x-x_0)^n
\end{equation}
such that $|x-x_0|<r$, then
...
0
votes
1answer
95 views
Limit of the determinant of a series of matrices
Given the $N\times N$ matrix $A$, consider the series:
$$B=\sum_{k=1}^{N}(A^k)^{-1}$$ where the symbol $o^{-1}$ means the inverse of $A^k$
is it possible and if yes how, to find all the matrices for ...
1
vote
1answer
220 views
The coefficients in the inverse of unit lower triangular Toeplitz matrix
I have a question about the inverse of lower triangular Toeplitz matrix.
There is a matrix $Q$ which can be infinite-dimensional.
$$
Q=\left[\begin{array}{ccccc}
1\\
-k_{1} & 1\\
-k_{2} & ...
0
votes
1answer
201 views
Calculating powers of 2 on a 2D grid without factoring.
Consider the following 2D infinitely large grid where the dots represent infinity:
...
1
vote
1answer
237 views
power series expansion of the square root of a Hermitian matrix
Is there a power series expansion of the square root of a Hermitian matrix, as a procedure to calculate the square root without taking the inverse or diagonalizing the matrix? I find for scalar number ...
4
votes
3answers
124 views
Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?
The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
2
votes
1answer
98 views
Determinant of a series of Hadamard matrix
Given: $$H=\ \left[ \begin{array}{cc|r}
1 & 1 \\ 1 & -1 \end{array} \right]$$
a Hadamard $H_2$ matrix.
and the series:
$$S=\sum_{k=0}^{N}{\frac{H^k}{k!}}$$
Is it possible to calculate ...
0
votes
2answers
623 views
Looking for a proof for the convergence of matrix geometric series
Consider a symmetric matrix $A$ with non-negative integer coefficients. It appears that the geometric series $\sum_{i \geq 0}A^i$ will converge to a matrix $B$ if the spectral radius (the largest ...
7
votes
2answers
276 views
Summing Matrix Series
I need to sum the series
$$I + A + A^2 + \ldots$$
for the matrix
$$A = \left(\begin{array}{rr}
0 & \epsilon \\
-\epsilon & 0
\end{array}\right)$$
and $\epsilon$ small. The goal is to ...
