Questions tagged [minimal-polynomials]

This is the lowest order monic polynomial satisfied by an object, such as a matrix or an algebraic element over a field.

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Prove or disprove if $m(x)=\prod_{j=1}^{s}(x-\lambda_{j}^{2})$ is the minimal polynomial of a linear map $T\circ T:V\to V$

Prove or disprove: If $m(x)=\prod_{j=1}^{s}(x-\lambda_{j}^{2})$ (with $\lambda_i\not=\lambda_j$ for $i\not=j$) is the minimal polynomial of a linear transformation $T\circ T:V\to V$, with $V$ a $\...
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Example for Matrix with $A^3=A$ that is not diagonalizable

I am working on an exercise where I am supposed to prove of disprove that $A\in\mathbb K^{n\times n}$ with $A^3=A$ is diagonalizable. My work: $$A^3=A\iff A^2(1-A)=0\Rightarrow \mu_A(X)=X\ \vee\ \mu_A(...
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The primary components of a space relative to a linear operator.

I am trying to understand a proof from N. Jacobson book ``Lectures in Abstract Algebra, II. Linear Algebra'', Chapter IV.8 (p.130, but I've attached the screen for convinience). Let $V$ be a finite-...
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Let $K/F$ and $a,b \in K$ algebraic over $F$ and $[F(a):F]=m ,[F(b):F]=n$ then show that degree of $a + b, ab, a − b , ab^{−1}$ atmost $mn$ over $F$

Let $K$ be a field extension of the field $F$ and $a,b \in K$ be algebraic over $F$.If a has degree $m$ over $F$ and $b \neq 0$ has degree $n$ over $F$, then show that the elements $a + b, ab, a − b , ...
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Fundamental Thm of Galois Theory: Showing that $L^H$ is normal over $F$ iff $H$ is normal in $G$

I am trying to understand the proof of the following Lemma (from this source) preceding the proof of the Fundamental Theorem of Galois Theory: We assume L/F to be a finite-dimensional Galois ...
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Showing that minimal polynomial of rank $r$ matrix has at most degree $r + 1$.

My attempt: Suppose the degree was $>r + 1$, i.e. $\mathbf{I}, \mathbf{A}, \ldots , \mathbf{A}^{r + 1}$ are linearly independent. This would mean \begin{align*} \forall \alpha _{0}, \ldots , \...
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Minimal polynomial of some elements in finite fields. [duplicate]

Describing Galois groups of some local fields (Ignore this quoted question, it is not related/ important) Following my question, I have some questions in my mind. Consider this special case: Let $p$ ...
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Unique matrix with minimal polynomial

Can anybody prove this. Up to similarity, there is a unique 3 × 3 matrix with minimal polynomial $(𝑥 − 1)^2(𝑥 − 2)$
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Is $p(\sqrt2)=2$, where $p$ is the minimal polynomial of $\sqrt{2}+\sqrt{-2}$?

Let $p(x)$ be the minimal polynomial of $\sqrt 2 +\sqrt{−2}$ over the field $\mathbb{Q}$ of rational numbers. Evaluate $p(\sqrt2)$. My attempt: Take $x= \sqrt 2 +\sqrt{−2}$, which implies $x-\sqrt ...
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Determine the basis of a composite field extension using the primitive element

I have been confusing myself a lot with the following and I am sure I must be missing something obvious, so sorry for this probably stupid question. Given $\alpha = \sqrt{2} - \sqrt{3}$ and $\beta = \...
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How to solve the $p$th power of matrix $C$?

if $p\equiv1(mod3)$ , $C=\left[ \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & -3 \\ 0 & 1 & -3 \\ \end{array} \right]\in M_3(\mathbb{F}_p).$ How to calculate $C^p$? ...
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Does the minimal polynomial and characteristic polynomial have same roots over F, for a linear operator on vectorspace V over the field F?

Actually my question is that whether the minimal polynomial and the characteristic have the same root over the field of the vectorspace or do they have the same root over any extension field of F. For ...
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minimal and characteristic polynomial of this operator [duplicate]

The following is Problem 18 from Chap8.C of Axler's Linear Algebra Done Right. Edited to add a transcription of the original problem(in the image) P18. Suppose $a_0, a_1, ...., a_{n-1} \in \mathbb{C}...
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Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some integer $m.$ Prove that $f^n = 0.$

Here is the question I want to tackle: Let $k$ be a field and let $V$ be an $n$-dimensional vector space over $k.$ Suppose that $f: V \to V$ is a $k$-linear transformation such that $f^m = 0$ for some ...
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Equality of minimal and characteristic polynomial [duplicate]

I'm trying to prove that for a companion matrix $C$ of a monic polynomial $f$, the minimal and the characteristic polynomial is the same. I am attempting a proof by Induction on the degree of $f$ but ...
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What is the number of linear independent eigenvectors of a complex matrix when the characteristic and minimal polynomials are the same? [closed]

Let $A$ be an $n \times n$ matrix with entries in $\mathbb{C}$. Suppose that the characteristic polynomial of $A$ is the same as the minimal polynomial of $A$ such that $$p_{A}(t) = m_{A}(t) = (t - \...
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Calculating Jordan Form of matrix - what to do with $rank(A^2)$?

I am told $m(x)$ divides $x^2(x-1)$, $rank(A) = n-1$ & $rank(A^2) = n-2$. So I have the base form: $$J_2(0)^a \oplus J_1(0)^b\oplus J_1(1)^c$$So I know the geometric multiplicity of the $0$ ...
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Minimal polynomial of $L_B$

Let $B$ be a matrix of the vector space $\mathcal{M}_{n\times n}(\mathbb{F})$, where $\mathbb{F}$ is a field, and let $L_B:\mathcal{M}_{n\times n}(\mathbb{F})\to \mathcal{M}_{n\times n}(\mathbb{F})$ ...
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Minimal polynomial of the matrix $A = \begin{bmatrix} c & 1 & 0 & 0\\ 0 & c & 0 & 0 \\ 0 & 0 & c & 1\\ 0 & 0 & 0 & c \end{bmatrix}$

I am learning about the minimal polynomial of a matrix for the first time, but I don't understand how to find quickly the minimal polynomial of some matrices. For instance, I know that the minimal ...
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1 answer
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Do we always have $\chi_{K/Q,x}=\pi_x$ if $K=\mathbb Q(x)$?

If $K=\mathbb Q(x)$ for $x\in\mathbb C$. Do we always have $\chi_{K/Q,x}=\pi_x$ the minimal polynomial of $x$? I am using the following definition: $\chi_{K/Q,x}$ is the characteristic polynomial of ...
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2 answers
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finding the minimal polynomial without using the characteristic polynomial

I want to find the minimal polynomial (the monic polynomial of least positive degree that annihilates the matrix) of the following matrix: $$\begin{pmatrix} 0 & 1 & 1 & 0\\ -1 & 0 &...
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Is it possible to find the minimal polynomial of a matrix knowing its rank?

We know that any $n \times n$ matrix $X$ with $rank(X)=1$ can be written $u(^Tv)$ where $u, v$ are vectors and $^TA$ is the transpose of $A$. This leads to $X^2=u(^Tv)u(^Tv)=u(^Tvu)(^Tv)=cu(^Tv)$, ...
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Minimal polynomial always equals minimal annihilating polynomial of some vector

Let $T:V\to V$ be a linear map. let $m_T$ be its minimal polynomial and $m_{T,x}$ be the minimal (degree monic) polynomial such that $m_{T,x}(T).x=0$ for $x\in V$. Question. Is it possible to show $...
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$5 \times 5$ nilpotent matrices with the same minimal polynomial and nullity must be similar.

Problem. Suppose $A, B$ are both $5 \times 5$ nilpotent complex matrices with the same nullity and the same minimal polynomial. Prove that $A$ and $B$ are similar. My Question. Is there a 'clever' ...
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"Linear algebraic" proof of Frobenius normal form

Theorem: Let $\mathcal{A}$ be a linear operator on a finite dimensional vector space $V$, there exists a basis such that $V$ can be represented by the direct sum of some $\mathcal{A}$-cyclic subspace ...
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Minimal polynomial of polynomial quotient field linear transfomation over finite field

$ \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\VP}{V.P.} %\DeclareMathOperator{\CCC}{C^1} \DeclareMathOperator{\contt}{C} \DeclareMathOperator{\PCC}{PC} \...
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1 answer
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Find for what $n$ and for what field $K$ exists $A \in GL_n(K)$ with order $n+1$

Find conditions on $n\in\mathbb{N}$ and $K$ field such that exists $A \in GL_n(K)$ with order $n+1$. Specify $A$. Explain why the same argument wouldn't work without those conditions. This is what I'...
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Algebraic Integers in Cyclotomic fields

Let $\zeta$ be a primitive 8th root of unity in $\mathbb{C}$ and let $\alpha = \frac{1-\zeta}{2}$ (i) Determine the minimal polynomial of $\alpha$ over $\mathbb{Q}$ (ii) Is $\alpha^{-1}$ an algebraic ...
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Triangularization of Matrix (theorem 5, chapter 6, Hoffman & Kunze)

My question pertains to the triangularization theorem, Chapter 6, Section 4, Theorem 5 in Hoffman & Kunze (HK) Linear Algebra. My question in short, if you are already familiar with the text: When ...
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Does multiplicity in minimal polynomial have a name?

Suppose operator $T$ on a finite dimensional vector space over algebraically complete field has characteristic polynomial $p(x) = (x - \lambda_1)^{r_1}(x - \lambda_2)^{r_2} \cdots (x - \lambda_k)^{r_k}...
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1 vote
2 answers
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Jordan canonical form of $p(\alpha)$ in terms of that of $\alpha$

Let $\alpha$ be a linear transformation defined in a finite-dimensional vector space $V$ over a field $F$. If polynomials $p(x)\in F[x]$ are such that for all eigenvalues $\lambda$ of $\alpha$, $p'(\...
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A problem about cyclic subspaces and minimal polynomial

Let $\alpha$ be a linear operator on a vector space $V$, and supoose that $V$ is $\alpha$-cyclic, say generated by $v\in V$. Suppose further that $V=U_1\bigoplus U_2$ for non-trivial $\alpha$-...
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An application of Zorn's Lemma in linear algebra

Let $\alpha$ be a linear operator on a vector space $V$. Let $v_0\in V$ and suppose that the minimal polynomials of $\alpha$ is $m_\alpha(x)$ and of $\alpha$ at $v_0$ is $m_{\alpha,\ v_0}(x)$. Suppose ...
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1 answer
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Columns & diagonalizability of a matrix.

I want to solve this question: Let $A$ be an $n \times n$ matrix over a field $k,$ all of whose columns are the same. Describe the conditions under which $A$ is diagonalizable. Justify your answer. I ...
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Systematic way to create Matrix with given minimal polynomial [duplicate]

Suppose you're given a minimal polynomial, is there an efficient and systematic (!) way to find a matrix with given minimal polynomial? For example for $n \in \mathbb{N}$ find an $n \times n$-Matrix ...
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Minimal polynomials in a tower of field extensions

Let $F\subset E\subset K$. Let $\alpha\in K$ be algebraic both over $F$ and $E$. I want to show that $\alpha\in K/F$ be separable means $\alpha\in K/E$ is separable. As an intermediate result, I'm ...
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When is it possible to find an invertible matrix of order $n+1$

Let $n$ be a natural and $K$ a field. When can we find an $n\times n$ invertible matrix $A$ of order $n+1$? Order here means $A^{n+1}=I$ and is the smallest integer that satisfies it. This is what I ...
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0 answers
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Determining the dimension of the Eigenspace without knowing the eigenvalues (Segre Classification)

This has been a burning problem in my head for a while now. Any help/suggestions are greatly appreciated. I'll use a concrete 3x3 matrix as an example, but I'd like to know thoughts especially for 4x4 ...
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0 answers
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Why does this polynomial splits over $\Bbb{F}_2(\sqrt{X})$

I have the following problem: Let $k=\Bbb{F}_2(X)$ and $E=\Bbb{F}_2(\sqrt{X})$. Then I know that the minimal polynomial of $\sqrt{X}$ over $k$ is $t^2-X$. But now in the lecture our prof. says that ...
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Given $\Delta_{T}(x)=m_{T}(x)=(x-\alpha)^{n}$ and $ TS = ST $ where $T,S$ are linear maps. Prove there is a polynomial $f(x)$ such that $S=f(T)$.

Problem: Let $V$ denote a vector space over $\mathbb{C}$. Let $T: V \rightarrow V$ denote a linear map such that $\Delta_{T}(x)=m_{T}(x)=(x-\alpha)^{n}$ for some $\alpha \in \mathbb{C} .$ Let $S: V \...
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1 vote
1 answer
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Properties of minimal polynomial with respect to the field elements

Let F be a field and V be a vector space on F. Let $m_\alpha$ be the minimal polynomial of linear operator $\alpha$ on $V$, and $deg\ (m_\alpha)=n$. Show that for each $\lambda_i\in F$ such that $m_\...
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1 vote
1 answer
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Seeing if a polynomial is irreducible over a field extension of $\Bbb Q$.

Exercise. Determine $m_{i\sqrt[3]{2}}$ over $\Bbb Q(i)$, where $m_{i\sqrt[3]{2}}$ represents the minimal polynomial of ${i\sqrt[3]{2}}$. My attempt. We know that $m_i = x^2 +1$ over $\Bbb Q$ (this is ...
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1 answer
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Proving that the smallest dimension for $V$ is 4.

Let $V$ a $\mathbb{Q}$-linear space, $\dim_\mathbb{Q}V<\infty$, $T: V \rightarrow V$ a linear operator such that $T^2 = -Id$. If $V$ has a $T$-invariant proper subspace $W$, $\dim(W) \ge 1$, then ...
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2 answers
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Why are these statements about separability equivalent?

I am reading something about separable extensions and passed by the following definitions: (Separable degree) Let $E$ be an algebraic extension of a field $F$. and let $\sigma :F\rightarrow L$ be an ...
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Show that the minimal polynomial is $M_{f}(t)=(t-\lambda_{1})^{d_{1}}...(t-\lambda_{k})^{d_{k}}$, $d_{i}$ is$(f|_{V_{i}}-\lambda_{i}I_{n})^{d_{i}}=0$.

I would like to show that the minimal polynomial is given by $M_{f}(t)=(t-\lambda_{1})^{d_{1}}...(t-\lambda_{k})^{d_{k}}$. I use the following definition of the minimal polynomial: Def.(Minimal ...
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1 vote
1 answer
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Concerning a minimal polynomial

I'm currently taking linear algebra and I'm working on a questing concerning minimal polynomial. The question is as follows: Consider a matrix $A$, whose minimal polynomial is of the form $p(x)^n$ ...
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1 vote
1 answer
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Characteristic polynomial of projection

Let $V$ be a $n$ dimensional vector space over a field $\mathbb{k}$, and let $P:V \rightarrow V$ be a linear map such that $P^2=P$. i.e., A linear map is a projection. I want to find all possible ...
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Define $T:M_{n}(\mathbf{C})\to M_{n}(\mathbf{C})$ by $T(A)=\frac{1}{2}(3 A+A^{t})$. Compute $m_{T}(x)$ and write a basis for all eigenspaces of $T$.

Problem: Define $T: M_{n \times n}(\mathbf{C}) \to M_{n \times n}(\mathbf{C})$ by $T(A)=\frac{1}{2}(3 A+A^{t})$. Compute $m_{T}(x)$ and write down a basis for all the eigenspaces of $T$. Attempt: From ...
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1 answer
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Minimal polynomial is lcm of minimal polynomials of invariant subspaces.

Question. I am working on Hoffman and Kunze, page 219, question 4c, they ask: Let $T$ be a linear operator on $V$. Suppose $V = W_1 \oplus \cdots \oplus W_k$ where each $W_i$ is invariant under $T$. ...
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3 votes
1 answer
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Assume that $A$ has the minimal polynomial as $(x-1)(x-2)$ and c.p. as $(x-2)^2(x-1)$

Let $$A := \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1\\ \end {pmatrix}$$ Find the number of matrices similar to $A$ whose entries are from $\mathbb{Z}/\mathbb{3Z}$. ...
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