List questions from companion-matrices

Published on 26 Mar 2026 - 4:53

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Companion matrix of matrix valued polynomial coefficients

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Existence of a real $4 \times 4$ matrix satisfying the equation $x^2+1=0$.

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Solving a general ODE

Published on 30 Apr 2026 - 9:52

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Determinant of sort of companion matrix

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30

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Generating synthetic multivariate time series with stable VAR model

Published on 09 Apr 2026 - 7:29

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Let $f \in \mathbb{F}[x]$ and $A_f=c(f) $ be the companion matrix of $f$. Given $ g \in \mathbb{F}[x] $ show that $ \dim \ker g(A_f) \leq \deg g $.

Published on 26 Mar 2026 - 3:13

160

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Companion matrices over the Galois field with two elements

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131

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Converting a monic polynomial of degree $n$ to its companion matrix

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If $b$ is a cyclic vector, what is significant about $\begin{bmatrix} A - \lambda I & b \end{bmatrix}$?

Published on 09 Apr 2026 - 13:06

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minimal and characteristic polynomial of this operator

Published on 26 Mar 2026 - 3:11

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Comparing values and signs of determinant of companion matrices.

Published on 26 Mar 2026 - 3:12

85

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Characteristic polynomial of A and -A (where A is a companion matrix)

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225

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Companion matrix must have one Jordan block per eigenvalue: recurrence sequence perspective

Published on 26 Mar 2026 - 4:50

57

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How to calculate the eigenvalues of $A^\ast A$ where $A$ is a companion matrix?

Published on 26 Mar 2026 - 4:53

List Question

Characteristic polynomials of powers of a companion matrix in terms of coefficients.

Companion matrix of matrix valued polynomial coefficients

Existence of a real $4 \times 4$ matrix satisfying the equation $x^2+1=0$.

Solving a general ODE

Determinant of sort of companion matrix

Generating synthetic multivariate time series with stable VAR model

Let $f \in \mathbb{F}[x]$ and $A_f=c(f) $ be the companion matrix of $f$. Given $ g \in \mathbb{F}[x] $ show that $ \dim \ker g(A_f) \leq \deg g $.

Companion matrices over the Galois field with two elements

Converting a monic polynomial of degree $n$ to its companion matrix

If $b$ is a cyclic vector, what is significant about $\begin{bmatrix} A - \lambda I & b \end{bmatrix}$?

minimal and characteristic polynomial of this operator

Comparing values and signs of determinant of companion matrices.

Characteristic polynomial of A and -A (where A is a companion matrix)

Companion matrix must have one Jordan block per eigenvalue: recurrence sequence perspective

How to calculate the eigenvalues of $A^\ast A$ where $A$ is a companion matrix?

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