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Results tagged with polynomials
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user 62673
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
4
votes
1
answer
152
views
Perturbing the constant term of a polynomial and implications to stability
Let $p(s)\in\mathbb{R}[s]$ be s.t.
$p(0)=0$;
$p(s)$ has at least one root in the right half complex plane $\{s\in\mathbb{C}\,:\,\Re\mathrm{e}(s)>0 \}$.
Then for every $\varepsilon\in\mathbb{R}$,
$ …
- 2,712
2
votes
0
answers
50
views
Metrics on the group of unimodular polynomial matrices
The group of unimodular matrices $\mathbb{U}[s]^{n\times n}$ is given by the set of $n\times n$ square (real) matrix-valued polynomials $\mathbb{R}[s]^{n\times n}$ which admit a polynomial inverse. … Equivalently, $\mathbb{U}[s]^{n\times n}$ coincides with the group of $n\times n$ square (real) matrix-valued polynomials that have non-zero constant determinant. …
- 2,712
5
votes
1
answer
152
views
Finding a particular matrix factor
Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$
$$
A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.
$$
I'm interested in finding a factorizat …
- 2,712
1
vote
0
answers
30
views
Eigenvalue assignment via state feedback: existence proof
Consider the linear time invariant system:
$$\tag{1}\label{eq1}
\dot{x}(t) = Ax(t) + Bu(t), \ \ x(0)=x_0\in\mathbb{R}^n,
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$. Let $p_M(s …
- 2,712