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Questions tagged [companion-matrices]

On the companion matrix to a certain monic polynomial.

The companion matrix of the monic polynomial $p(t)=c_0+c_1 t+\cdots+c_{n-1} t^{n-1}+t^n,$ is the square matrix defined as

$C(p)=\begin{bmatrix}0&0&\cdots&0&-c_0\\1&0&\cdots&0&-c_1\\0&1&\cdots&0&-c_2\\\vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\cdots&1&-c_{n-1}\end{bmatrix}$.

The characteristic polynomial as well as the minimal polynomial of $C(p)$ are equal to $p$.

In this sense, the matrix $C(p)$ is the "companion" of the polynomial $p$.

Companion matrices are too related with the Rational Canonical Form of a matrix.

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The characteristic and minimal polynomials of a companion matrix

The companion matrix of a monic polynomial $f \in \mathbb F\left[x\right]$ in $1$ variable $x$ over a field $\mathbb F$ plays an important role in understanding the structure of finite dimensional $\mathbb F[x]$-modules. It is an important fact that…
DBr
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Determinant of a companion matrix

Find determinant of $$ A := \begin{bmatrix} 0 & 0 & 0 & \dots & 0 & a_0 \\ -1 & 0 & 0 & \dots & 0 & a_1 \\ 0 & -1 & 0 & \dots & 0 & a_2 \\ 0 & 0 &…
user8277998
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Characteristic polynomial of companion matrix

I have a matrix in companion form, $$A=\begin{pmatrix} 0 & \cdots & 0& -a_{0} \\ 1 & \cdots & 0 & -a_{1}\\ \vdots &\ddots & \vdots &\vdots \\ 0 &\cdots & 1 & -a_{n-1} \end{pmatrix}$$ where $A \in M_{n}$. I want to prove by induction that the…
Edison
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Find the roots of a polynomial using its companion matrix

I would like to find the roots of a polynomial using its companion matrix. The polynomial is ${p(x) = x^4-10x^2+9}$ The companion matrix $M$ is $M={\left[ \begin{array}{cccc} 0 & 0 & 0 & -9 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 10 \\ 0 & 0 & 1 &…
user60736
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Is it possible to find a companion matrix of a polynomial which is also hermitian?

The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial $$ a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~, $$ I can define a companion matrix $$ A[p]=\begin{bmatrix} 0…
sintetico
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Eigenspace of the companion matrix of a monic polynomial

How do I prove that the eigenspace of an $n\times n$ companion matrix $$ C_p=\begin{bmatrix} 0 & 1 & 0 &\cdots & 0\\ 0 & 0 & 1 &\cdots & 0 \\ \vdots&\vdots &\vdots&\ddots&\vdots\\ 0 & 0 & 0 &\cdots &1 \\ -\alpha_0 &-\alpha_1 &-\alpha_2…
L1meta
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Are there general methods to compare the largest real roots of two integer-coefficient polynomials using only their expressions?

Question Are there general methods to compare the largest real roots of two integer-coefficient polynomials using only their expressions? Example 1: Consider: $P_{1}\left ( x \right )= x^{5}- 20x^{4}+ 147x^{3}- 505x^{2}+ 784x- 432$ $P_{2}\left (…
Dang Dang
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Companion matrix of bivariate polynomial

A polynomial in one variable can be expressed as a companion matrix, of which the eigenvalues are the roots of the polynomial and which can be found by using e.g. QR decomposition or power iteration. Is there anything like this for multivariate…
Ecir Hana
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Eigenvalues of companion matrix of $4x^3 - 3x^2 + 9x - 1$

I want to find all the roots of a polynomial and decided to compute the eigenvalues of its companion matrix. How do I do that? For example, if I have this polynomial: $4x^3 - 3x^2 + 9x - 1$, I compute the companion matrix: $$\begin{bmatrix}…
rubik
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How to verify this matrix identity?

Let $m\geq 3$ be a positive odd number and let $M$ be the $m\times m$ matrix defined by $$M=\begin{bmatrix}0&1&0&0&\cdots&0\\ 0&0&1&0&\cdots &0\\ 0&0&0&1&\cdots &0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&\cdots…
Zuriel
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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 form: we first construct a block diagonal matrix…
user1101010
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Is the characteristic equation in ODE the same characteristic equation in linear algebra?

Can someone show me whether this "characteristic equation" thing in ODE is the same characteristic equation that we derive for a matrix? For example, given $y'' + 2y = 0$, the characteristic equation is $\lambda^2 = -2$ How does this equation…
Olórin
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Show each eigenvalue of a companion matrix has geometric multiplicity $=1$.

Given the differential equation $$x^{(n)}(t)+c_{n-1}x^{(n-1)}(t) + \dotsb + c_1x'(t) + c_0=0,$$ we can form a vector $\xi = (x, x', \dotsc, x^{(n-1)})$, and then we have $$\xi'(t) = A\xi,$$ where $A$ is the transpose of the companion matrix for the…
Eric Auld
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How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?

A few days ago, I had a vague question in my mind about "matrix methods" for finding the roots of a polynomial. Now, I can ask at least a semi-precise question, thanks to the post How to calculate complex roots of a polynomial. I do not have a…
Srivatsan
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Quick way of showing an $n\times n$ Jordan block associated to $1$ is similar to the companion matrix of $(x-1)^n$

Is there a quick, clean way of proving that the $n\times n$ Jordan block with $1$'s on the diagonal and the Frobenius companion matrix corresponding to the polynomial $(x-1)^n$ are similar matrices? Apparently, the (triangular) Pascal matrix is the…
Joost
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