# Questions tagged [companion-matrices]

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### When are almost block companion matrices which yield a given characteristic polynomial connected?

This is motivated by this question Are matrices which yield a given characteristic polynomial and have specified structure connected? Let $\mathcal E \in M_n(\mathbb R)$ be a subset with following ...
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### Does every polynomial with a Perron root have a primitive matrix representation?

Let $p(x)=x^6-13x^4-20x^3+x^2-x+2$ and $C$ be the companion matrix of $p(x)$. How can I find a primitive matrix similar to $C$ ? Is there a general method to transform the companion matrix with a ...
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### How to check a special form of Companion Matrix is a Primitive Matrix or not?

Definition: A non-negative matrix square $A$ is called primitive if there is a $k$ such that all the entries of $A^k$ are positive. Lemma: If the graph associated to $A$ is strongly connected and ...
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### Proof of FLOPS of matrix inverse by companion matrix

I was wondering if anyone could help me prove that the cost of matrix inversion by companion matrix is $2n^3$ + L.O.T. FLOPS.
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### Are companion matrices useful for binary division?

The new find-first-set bit "ffs" CPU instruction found in the multi media extensions (MMX) 4 apparently made possible to start doing Newton-Raphson division (according to Wikipedia). Does someone ...
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### Polynomials $\&$ Matrices
Assume $A$ is a matrix of order $n$. We know that the characteristic polynomial of matrix $A$ is obtained as follows $$P(x)=\det (A-x\,I)\, .$$ Where $I$ is an identity matrix of order $n$. What ...