1
vote
2answers
49 views

Matrix Inversion Test ( Sum of Matrix series)

Friends,I have a set of matrices of dimension $3\times3$ called $A_i$. , Following are the given conditions a) each $A_i$ is non invertible except $A_0$ because their determinant is zero. b) ...
2
votes
1answer
42 views

power series for matrix with elements smaller than 1

If I have a square matrix A such that all elements $|a_{ij}| < 1$ does this guarantee that all my eigenvalues will also be less than 1 and that the power series $S = I - A + A^2 - A^3...$ will ...
2
votes
2answers
111 views

Given a perturbation of a symmetric matrix, find an expansion for the eigenvalues

Let $A$ be a real, symmetrix $n\times n$ matrix with $n$ distinct, non-zero eigenvalues, and let $V$ be a real, symmetric $n\times n$ matrix. Consider $A_{\varepsilon}=A+\varepsilon V$, a ...
0
votes
0answers
79 views

Algorithm for reversion of power series?

Is this an algorithm for power series reversion? As input I give the alternating reciprocals of the factorial numbers: ...
2
votes
1answer
34 views

Are power series in a normal matrix themselves normal?

Are (convergent) power series in a normal matrix themselves normal? I have looked around for this result, and not found it. How might we prove it?
1
vote
0answers
127 views

Power series expansion of the minimum eigenvalue of a linear matrix function

Let $M(\alpha) = A + \alpha \, B$, where A, B are two $n\times n$ positive-semidefinite matrices, and $\alpha$ is a scalar. Define $\lambda(\alpha)$ as the smallest eigenvalue of $M(\alpha)$, i.e., $$ ...
1
vote
1answer
54 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
102 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
152 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
196 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
425 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
253 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
389 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 ...
5
votes
3answers
141 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
107 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
804 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
394 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 ...