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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Is the characteristic polynomial of a minimal linear recurrence sequence also the minimal polynomial?
Let $\{u_m\}_{m=0}^\infty$ be a sequence of terms satisfying a linear recurrence relation with rational coefficients of order $k$, so that $\{u_m\}_{m=0}^\infty$ does not satisfy a recurrence relation ...
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kernel of $K[X,Y] \rightarrow K[\theta_1, \theta_2]$
Given a field $K$, for example the field of rational numbers ${\bf Q}$, and two algebraic numbers $\theta_1$, $\theta_2$ over $K$, what is the kernel of the morphism $$K[X,Y] \rightarrow K[\theta_1, \...
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The author says it is clear that $(A^{(i)}-\alpha I_{n_i'})^{\nu_i-1}\neq 0$. But it is not clear for me. ("Linear Algebra" by Ichiro Satake.)
I am reading "Linear Algebra" by Ichiro Satake.
The author says it is clear that $(A^{(i)}-\alpha I_{n_i'})^{\nu_i-1}\neq 0$.
But for me it is not clear.
Let $\mathbb{C}^n$ be an $n$-...
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Infinite dimensional analogue of Minimal Polynomial
I was wondering if an analogue of the minimal polynomial diagonalization theorem exists in infinite dimensions, in particular I was thinking : suppose we have a bounded linear operator $T: H \to H$ ...
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$f_A(x)$ is the characteristic polynomial of $A$. $\phi_A(x)$ is the minimal polynomial of $A$. $f_A(x)/\phi_(x)$ has some property. (Ichiro Satake.)
I am reading "Linear Algebra" by Ichiro Satake.
I want to know how to prove the fact that the author wrote in Remark below.
My attempt:
$f_A(x)E=(xE-A)B(x)=(xE-A)\psi(x)B'(x)$.
$\phi_A(x)E=(...
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Intuition behind degree of minimal polynomial
I'm learning linear algebra from Linear Algebra Done Right. The book gives a proof that the degree of the minimal polynomial of an operator on V is at most the dimension of V, but the proof feels ...
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Necessary and sufficient condition to have an upper-triangular matrix
The following is result 5.44 from Linear Algebra Done Right, fourth edition by Sheldon Axler:
Theorem. Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Then $T$ has an upper-triangular ...
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Prove that the minimal polynomial of $T_{\mathbb{C}}$ equals the minimal polynomial of $T$.
This is exercise 5.B.27 in Linear Algebra Done Right, fourth edition by Sheldon Axler:
Suppose $\mathbb{F} = \mathbb{R}$, $V$ is finite-dimensional, and $T \in \mathcal{L}(V)$. Prove that the minimal ...
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Minimal polynomial of $T / U$ times minimal polynomial of $T|_U$ is a polynomial multiple of the minimal polynomial of $T$
The following is exercise 5.B.25 form Linear Algebra Done Right, fourth edition by Sheldon Axler (https://linear.axler.net/LADR4e.pdf)
Suppose $V$ is finite-dimensional, $T \in \mathcal{L}(V)$, and $...
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Exercise 5.B.19 from "Linear Algebra Done Right", Sheldon Axler, 4th edition.
The following is exercise 5.B.19 from Linear Algebra Done Right, Sheldon Axler, fourth edition.
Suppose $V$ is finite-dimensional and $T \in \mathcal{L}(V)$. Let $\mathcal{E}$ be the subspace of $\...
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Find the number of matrices over the finite field $\mathbb F_{19}$, whose minimal polynomial has a certain degree $m$.
I am collaborating with some colleagues to create a TACA (a test assessing knowledge in Calculus, Linear Algebra, and Elementary Group Theory) practice test. During this process, I devised the ...
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Finding BCH code syndromes
I' m not getting how syndromes are calculated for bch codes so I tried finding examples but still I don't seem to have it
To calculate the first syndrome for the received message polynomial
$R(x)=1+...
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Proof of: Connected graph with three distinct adjacency matrix eigenvalues with the largest eigenvalue nonintegral is complete bipartite.
In the paper "Graphs with Three Eigenvalues" by E. R. van Dam, proposition 2 says that if a connected graph has three distinct eigenvalues and the largest is nonintegral, then it's a ...
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Minimum polynomials of elements in purely inseparable extensions
I am trying to solve this exercise in field theory :
Assume that $k$ is a field of prime characteristic $p$, and assume that $L$ is a purely inseparable extension of $k$. Prove that the minimum ...
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Define $T \in \mathcal{L}(\mathbb{F}^n)$ by $T(x_1, x_2, x_3, \ldots, x_n) = (x_1,2x_2,3x_2,\ldots, nx_n)$. Find the minimal polynomial of $T$.
The following is an exercise in "Linear Algebra Done Right" by Sheldon Axler, 4th edition.
Define $T \in \mathcal{L}(\mathbb{F}^n)$ by $T(x_1, x_2, x_3, \ldots, x_n) = (x_1,2x_2,3x_2,\ldots,...
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Exercise 5.B.7(b) in "Linear Algebra Done Right" 4th edition by Sheldon Axler
The following exercise is part (b) of exercise number 7 in Sheldon Axler's Linear Algebra Done Right, 4th edition:
Suppose $V$ is finite-dimensional and $S, T \in \mathcal{L}(V)$. Prove that if at ...
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Does the constant term of the minimal polynomial of an algebraic function also have no univariate factor if the algebraic function has none?
Each algebraic function is defined by an irreducible algebraic equation or its minimal polynomial, respectively.
My question is:
Let
${}^{-}$ denote the algebraic closure,
$\pmb{n\in\mathbb{N}_{>1}...
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Determine the minimal polynomial of $T \in \mathcal{L}(\mathbb{F^n})$ defined by $ T(x_1, \ldots, x_n) = (\sum x_i , \ldots, \sum x_i)$
The following exercise come from Linear Algebra Done Right, 4th edition, Sheldon Axler (I removed parts of the exercise that do not pertain to my question).
Suppose $n$ is a positive integer and $T \...
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Linear operator annihilates vector for every polynomial
I am working on this problem. I am studying for an exam.
Let $T$ be a linear operator on a finite-dimensional vector space $V$. Show that there is a non-zero vector $v \in V$ such that for all $f \in \...
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Why is $x^2+x+1$ a factor of the minimal polynomial over $\Bbb R$ just because $x^2+x+1$ is a factor of the characteristic polynomial? [duplicate]
I was studying minimal polynomials in a Linear Algebra course. I am using the book Linear Algebra by Stephen H Friedberg, Insel, and Spence for this purpose.
I was doing a problem but while reading a ...
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Find minimal polynomial of a matrix with $a_{ij} = a$ for every $a_{ij}$ [duplicate]
I want to find the minimal polynomial of the following matrix:
$A = (a_{ij})\in M_n(K) $ with $ a_{ij} = a \not = 0$ for every i and j
I found the matrix diagonalizable as dim ImA = 1 and the matrix ...
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Find all possible Jordan forms of a real matrix $A$ of order 7 whose minimal and characteristic polynomial is $(x-2)^3(x+3)^2$ and $(x-2)^4(x+3)^3$
Our professor recently taught us minimal polynomial in a Linear Algebra course. At the end of his lecture, he wrote on board just how does a Jordan matrix looks like and told us to consider that as ...
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Block Matrices and minimal, characteristic polynomials
For context, I'm a second year undergraduate maths student, preparing for my exams.
Let $f(x) = x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + \dots + a_1 x + a_0$, and suppose $V$ is a finite-dimensional ...
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Prove that the minimal and the characteristic polynomial of a linear operator are the same
Suppose $V$ is an $n$ dimensional vector space over a field $F,$ and $B=\{v_1,v_2,...,v_n\}$ be an ordered basis. Let $T:V\to V$ be the linear operator such that $T(v_1)=v_2, T(v_2)=v_3,...,T(v_{n-1})=...
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Show that the minimal polynomial of a linear operator is same as the minimal polynomial of its matrix representation wrt any ordered basis.
Let $T$ be a linear operator on a finite-dimensional vector space $V,$ and let $β$ be an ordered basis for $V.$ Then the minimal polynomial of $T$ is the same as the minimal polynomial of $[T]_β.$
I ...
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Notable algebraic numbers with high minimal polynomial degree
In this question I'll be referring to certain numbers as "notable". To remove the possible objection of this being opinion-based, we may define "notable" to mean someone has ...
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Why the orthogonal projection give minimal?
Given the data $ \vec{x}=(-2,-1,1,2) )$ and $( y=(1,1,-1,1) )$. Use an orthogonal projection to determine the coefficients $( a_{0}, a_{1}, a_{2} )$ of the quadratic polynomial function
$
\begin{...
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Artin's theorem exercise - proving that the fixed field is generated by the coefficients of the minimal polynomial
Suppose $L/K$ is a finite extension. $G$ is a finite group of $K$-automorphisms of $L$. Denote by $L^G$ the field elements of $L$ fixed by action of $G$. For any $\alpha \in L$ we write $f(t, \alpha) =...
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Every operator on a finite-dimensional vector space of dim ≥ 2 has invariant subspace of dim=2
I'm attempting to find a solution for this problem, and have written a proof and am unsure of a few steps, which I will mark. This is for self-study, so I appreciate the feedback. I'm also wondering ...
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How to find a matrix, characteristic and minimal polynomial of a linear operator?
Given linear operator: $L : \mathbb{R}^3[X] → \mathbb{R}^3[X]$
$$L(g(X))=g(0)(-1-2X+4X^2)+g^`(1)(-1+2X^2)+g^{``}(0)(-X+\frac{1}{2}X^2)$$
Find the matrix of operator L, characteristic and minimal ...
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Minimal polynomial question (LADR, 4th Ed)
I'm reading the following question from Linear Algebra Done Right (Fourth Edition).
However, I must be misunderstanding something. Take $p$ as the minimal polynomial of T. Then $p(T) = 0$. However, ...
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$\operatorname{rank}(A)=r$, then $I,A,\cdots, A^{r+1}$ is linearly dependent
Let $A$ be a $n\times n$ matrix, $\operatorname{rank}(A)=r$, then $I,A,\cdots, A^{r+1}$ is linearly dependent.
I know just if the degree of the minimal polynomial of $A$ is $k$, then $I, A, \cdots, A^...
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Degree of field extension $\Bbb Q(\sum\limits_{k=1}^{\text{ord}_n(2)}\zeta_n^{2^k}):\Bbb Q$
$n>2$ is an odd squarefree integer.
Let $\zeta_n=\mathrm{e}^{i\frac{2\pi}n}$ be a primitive $n$-th root of unity.
Is it true that $[\Bbb Q(\sum\limits_{k=1}^{\text{ord}_n(2)}\zeta_n^{2^k}):\Bbb Q]=\...
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A reference for the definition of a minimal polynomial when ring is noncommutative
Let $R$ be a non-commutative ring and consider $R$ as a subring of $M_n (R)$ the ring of $n\times n$ matrices with entries in $R$. Let $A\in M_n (R)$. I was wondeing if someone could give a reference ...
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Classify, up to similarity, the $3$ by $3$ matrices with coefficients in $\mathbb{Q}$ that satisfy $A^6=I$.
I am working on the following question in review for my algebra final:
Classify, up to similarity, the $3$ by $3$ matrices with coefficients in $\mathbb{Q}$ that satisfy $A^6=I$.
My work:
As $A^6 - ...
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Minimal polynomial of a recurrent sequence
Let $u$ ${\in}$ ${F^{\infty} _{2}}$ be a linear recurrent sequence with an irreducible characteristic polynomial f(x). Find the minimal polynomial of the sequence $\bar{u}$, where
$\bar{u}$ is the ...
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Find minimal polynomial over $\mathbb{F}_p$ of a generator of $\mathbb{F}_{p^n}^{*}$
So this is part of an exercise sheet where I thought I had figured it out but turns out I didn't.
Theory from lecture:
We know for a prime $p$ the finite field $\mathbb{F}_{p^n}$ is isomorphic to $\...
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$ \lambda^{*}(n) $ minimal polynomials
I already asked a closely related question on MSE.
Considering the modular lambda function, the values of $ \lambda^{*}(n) $ for some integers are given on here.
Is there a way to calculate the ...
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find minimal polynomials
I have the following task:
Let $\alpha$ be a complex number satisfying $\alpha ^3 +2\alpha -1 =0$ find its minimal polynomial and the minimal polynomal of $\alpha^2 +\alpha$.
For the first part tried ...
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If $B^3=B$, is $B$ diagonalizable?
Let $B\in M_n(\mathbb{F})$ such that $B^3=B$. Is $B$ diagonalizable?
If $B^3=B$, then $B^3-B=0$. Consider the polynomial $p(x)=x^3-x$. If $p(B)= B^3-B=0$. Since we know that the minimal polynomial of ...
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Patterns in the minimal polynomial of sum of square roots
n = $1$: Let the root of $x+p=0$ be $x_1$.
The minimal polynomial of $\sqrt{x_1}$ is
$$(x-\sqrt{-p})(x+\sqrt{-p})=x^2+p$$
n = $2$: Let the roots of $x^2+px+q=0$ be $x_1,x_2$.
The minimal polynomial of ...
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Is there a unique irreducible polynomial relating two algebraic numbers beyond their minimal polynomials?
Given two algebraic numbers a,b over $\mathbb{Q}$, with $p(x) \in \mathbb{Z}[x]$ being the minimal polynomial of $a$, and $q(y) \in \mathbb{Z}[y]$ the minimal polynomial of $b$, suppose that there ...
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Let $a,b$ be algebraic over the field $F$, with $\deg(a,F)$ and $\deg(b,F)$ coprime. Is $\deg(a+b,F)=\deg(a,F)\deg(b,F)$? What about $\deg(ab,F)$?
Let $a,b$ be algebraic over the field $F$, such that $\deg(a,F)$ and $\deg(b,F)$ are coprime. Then is $\deg(a+b,F)=\deg(a,F)\deg(b,F)$? What about $\deg(ab,F)$?
I have tried to represent $a,b$ as ...
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Is the Jordan normal form uniquely determined by the characteristic and minimal polynomial if they are the same?
I'm studying for an exam and I can't get anywhere with a problem. I've seen similar questions on here but not the same.
The problem provides the characteristic polynomial $XA(x) = (x-3)^2(x-1)$ and ...
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Minimal polynomial degree formula?
Let $u_i$ be all the distinct zero's of its (irreducible) minimal polynomial $f(x)$ of degree $n$ :
$$ f(x) = x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... + a_n $$
where the coefficients $a_j$ are integers.
...
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$f$ is diagonalizable iff its minimal polynomial is "free from squares" (proof)
Let $f \in End(V)$; then $f$ is diagonalizable iff its minimal polynomial is "free from squares", as in, all of its terms are all raised to the first power and (edit:) all irreducible ...
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$A$-invariant subspace of $\mathbb{F}^n$
Let $\mathbb{F}$ be a field and $A\in M_n(\mathbb{F})$.
Let the minimal polynomial of $A$ be $\prod_{i=1}^m q_i(x)^{\ell_i}$, where each $q_i(x) \in \mathbb{F}[x]$ is monic irreducible, each $\ell_i&...
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Proving a canonical height property
Let $H$ be the classical Weil height for a number field (of degree $n$ say), i.e. given $\alpha$ an element of this field and $f(x) = a_0x^n + \cdots + a_n \in \mathbb{Z}[X]$ its minimal polynomial, ...
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Prove that $T-\lambda I$ has minimal polynomial $q(z)=p(z+\lambda)$
I am learning Linear Algebra Done Right. Here is a question from 5B.
Suppose $V$ is finite-dimensional, $T\in L(V)$, and $p$ is the minimal polynomial of T. Suppose $\lambda\in F$. Show that the ...
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The minimal polynomial splits over a field of prime characteristic?
The statement of the exercise from the textbook goes:
Let $a \in E$, where $E$ is an algebraic extension of a field $F $ of prime characteristic $p$. Let $m(X)$ be the minimal polynomial of $a$ over ...
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