# Tagged Questions

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### Condition for linear minimal polynomials

I'm just wondering that there is a necessary and sufficient condition for minimal polynomials for in which cases are them linear. Let $A$ be a square matrix. I think that $A$ has a linear minimal ...
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### Show that $p(A) = \begin{bmatrix} p(A_{11})&(Mess)\\ 0&p(A_{22})\\ \end{bmatrix}$ for any polynomial $p(x)$. (See problem for full question.)

Important Note: This is a homework problem. The full question is as follows: If $$A = \begin{bmatrix} A_{11}&A_{12}\\ 0&A_{22}\\ \end{bmatrix}$$ where $A_{11}$ and $A_{22}$ are square, ...
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### If $p(x)=x^2-cx$ annihilates $A$, then $A$ is similar to $c \operatorname{diag}(1,\dots,1,0,\dots,0)$.

Let $A \in \mathbb{C}^{n \times n}$ and $0 \neq c \in \mathbb{C}$ a given constant. Suppose that $A$ has the following property: $A^2 =cA$. I had a question about this matrices, and I get an anwser, ...
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### Convex combination of polynomials with roots on the unit circle and companion matrix

Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ ...
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### prove that if gcd(f, minimal polyonimial of A) is not 1 then f(A) is not invertible

A is square matrix and f is polynomial. prove that if gcd(f, minimal polyonimial of A) is not 1 then f(A) is not invertible. any hints please..