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
0answers
49 views

How to power series expand determinants?

Say $g$ is a ($d\times d$) matrix which is given as, $g = g_0 + xg_2 + x^2 g_4 .. +x^{d/2 -1}g_{d-2}+ x^{d/2}(g_d + h_d(log (x)))$ where $d$ is an even number and each $g_i$ is a matrix (same ...
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 ...
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
2answers
157 views

How to denote the opposite case of the Kronecker Delta?

The Kronecker delta is defined as link to wikipedia: $$\delta_{l,m} = \begin{cases} 1 & \text{if }m=l,\\ 0 & \text{if }m\neq l. \end{cases}$$ I would like to denote the case where: $$ = ...
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
2answers
161 views

find a solution from power series for multiple variable

$3^x4^y = 4,782,969 $ where $x$ and $y$ are integers. What is the value of $y$? Is there any theory to solve this type problem? i have tried to make $4,782,969$ into power series but couldn't. So a ...
1
vote
3answers
84 views

power series of a matrix well-defined

I am working on a seminar lecture and have found the following lemma without a proof: Given a convergent power series $f(z)=\sum_{n=0}^\infty a_nz^n$ and a diagonalizable matrix $M$ with diagonal ...
0
votes
2answers
94 views

Power Series of Linear Transformations

In a course I'm taking, we're talking about polynomials of linear operators, i.e. if $E$ is a linear transformation on $V$ and $p$ is a polynomial then we can consider the linear transformation ...
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
9
votes
2answers
346 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
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} & ...
3
votes
1answer
83 views

Dimension of a quotient vector space of meromorphic functions

Let $U$ be an open set of the Riemann sphere, $z_i$ be $n$ distinct points of $U$, and $E$ the vector space of meromorphic functions on $U$ with poles of order no more than 2. Let $F$ be the subspace ...
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 ...
1
vote
1answer
174 views

Two equations with one solution over infinite variables

Apparently, my previous question didn't get no satisfactory answer, when I asked for two equations having a fixed value for each, not necessarily linear. As XenoGraff states, WolframAlpha does the ...
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 ...
0
votes
3answers
103 views

Geometric Series Converges? [duplicate]

Possible Duplicate: Value of $\sum\limits_n x^n$ If I have some real $x$ where $0 < x < 1$ What is the value $y = x + x^2 + x^3 + x^4 + \dots$ ? Intuitively I can see that for $x = ...
1
vote
2answers
318 views

Formal power series and vector (sub-)spaces

Let $$\mathbb{C}[[x]] := \{\sum_{n\geq 0} a_n x^n | a_n \in \mathbb{C}\}$$ be the set of formal power series of x and $V$ be the vector space of all series over $\mathbb{C}$. Let in addition $V_1$ be ...
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 ...
15
votes
2answers
925 views

any pattern here ? (revised 2)

for any positive number $k$, I have a $(k+1)*(k+1)$ matrix. I wonder if these matrices follow any "obvious" pattern. My goal is to guess the elements for matrix with $k=5$ and above (most probably in ...