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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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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