# Tagged Questions

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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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### 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 ...
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### 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 ...
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### 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 ...