# Tagged Questions

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### Recurrence - using power series

Could you help me in solving this recursion( a closed form ) using power series $\mu(n)=\mu(n−1)k_0+(n−1)\mu(n−2) k_1 \tag 1$, where $k_0,k_1$ are constants $\mu(0)=3,\mu(1)=5$ HINT: We can think ...
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### power series for square root matrix

Suppose I have a matrix of the form $$U\ =\ (I+z\thinspace X)^{\frac{1}{2}}$$ where $I$ is the $n\times n$ identity matrix, $z\in\mathbb{C}$ and $X$ is a $n\times n$ arbitrary complex matrix with ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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?
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### 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
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### 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 $$f(x)=\sum_{n=0}^{\infty}c_n(x-x_0)^n$$ such that $|x-x_0|<r$, then ...
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 ...