minimal polynomial Definition and 39 Threads

The usual theorem is talking about the linear operator being restricted to an invariant subspace: I had no problem understanding its proof, it appears here for example: https://math.stackexchange.com/questions/3386595/restriction-operator-t-w-is-diagonalizable-if-t-is-diagonalizable However, I...

Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be; ##α^2=5+2\sqrt{6}##

I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...

Hello! (Wave) If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$. ($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$) Is there a...

Homework Statement Let a = (1+(3)^1/2)^1/2. Find the minimal polynomial of a over Q. Homework EquationsThe Attempt at a Solution Maybe the first thing to realize is that Q(a):Q is probably going to be 4, in order to get rid of both of the square roots in the expression. I also suspect that...

Homework Statement So I need the find the minimal polynomial of the primitive 15th root of unity. Let's call this minimal polynomial m(x) Homework EquationsThe Attempt at a Solution I know that m(x) is an irreducible factor of x^15 - 1 and also that the degree of m(x) is equal to the Euler...

Homework Statement Find the minimal polynomial of a = i*(2)^1/2 + (3)^1/2 Homework EquationsThe Attempt at a Solution Well, I know the minimal polynomial will have degree four, and that's about it. Will it help if I look at the linear factors of the minimal polynomial in some splitting field...

Hey! :o We suppose that $M/L/K$ are consecutive fields extensions and $a\in M$ is algebraic over $K$. I want to show that $a$ is algebraic also over $L$. I want to show also that the minimal polynomial of $a$ over $L$ divides the minimal polynomial of $a$ over $K$ (if we consider this...

I started by setting $\alpha= e^{2\pi i/3} + \sqrt[3]{2}.$ Then I obtained $f(x) = x^9 - 9x^6 - 27x^3 - 27$ has $\alpha$ as a root. How can I proceed to find the minimal polynomial of $\alpha$ over $\mathbb{Q},$ and identify its other roots?

I'm trying find the minimal polynomial of $$a=3^{1/3}+9^{1/3}$$ over the rational numbers. I am currently going about this by trying to construct a polynomial from a (using what I intuitively feel would be a sufficiently small number of operations). Then I'd show it's irreducible by decomposing...

Homework Statement . Let ##X:=\{A \in \mathbb C^{n\times n} : rank(A)=1\}##. Determine a representative for each equivalence class, for the equivalence relation "similarity" in ##X##. The attempt at a solution. I am a pretty lost with this problem: I know that, thinking in terms of...

Homework Statement M: V -> V linear operator st M^2 + 1_v = 0 find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v Homework Equations The Attempt at a Solution using M^2 = -1_v, i rewrote the operator(?) as M^3 + M + I_v i don't know what to do. i guessed min poly to...

i understand how to find minimal poly. if a matrix is given. i am curious if you can find the matrix representation if minimal polynomial is given. i'm not exactly sure how you could since you can possibly lose repeated e-values when you write minimal polynomial. how can u create a n...

Hi everyone, :) This is one of the thoughts that I got after thinking about finding the minimal polynomial of a matrix. I know that the minimal polynomial is easy to find when a matrix is diagonalizable. Then the minimal polynomial only consist all the distinct linear factors of the...

Homework Statement Given the matrix 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 What is the minimal polynomial? Homework Equations - The Attempt at a Solution This is the Jordan form, so I guess the solution is just...

Let $L$ be an extension of a field $F$. Let $\alpha_1, \alpha_2\in L$ be such that both of them are algebraic over $F$ and have the same minimal polynomial $m$ over $F$. We know that there is an isomorphism $\phi:F(\alpha_1)\to F(\alpha_2)$ defined as $\phi(\alpha_1)=\alpha_2$ and $\phi(x)=x$...

Assume that \sqrt[M]{N} is irrational where N,M are positive integers. I'm under belief that X^M-N is the minimal polynomial of \sqrt[M]{N} in \mathbb{Q}[X], but I cannot figure out the proof. Assume as an antithesis that p(X)\in\mathbb{Q}[X] is the minimal polynomial such that \partial p <...

Homework Statement Given field extension C of Q, Find the minimal polynomial of a=sqrt( 5 + sqrt(23) ) (element of C).Homework Equations The Attempt at a Solution I may be complicating things, but let me know if you see something missing. Doing the appropriate algebra, I manipulated the above...

Let A ∈ Mn×n(F ) Why dim span(In, A, A2, A3, . . .) = deg(mA)?? where mA is the minimal polynomial of A. For span (In,A,A2...) I can prove its dimension <= n by CH Theorem but what's the relation between dim span(In,A,A2...)and deg(mA)

I've been given a matrix A and calculated the characteristic polynomial. Which is (1-λ)5. Given this how does one calculate the minimal polynomial? Also just to check, is it correct that the minimal polynomial is the monic polynomial with lowest degree that satisfies M(A)=0 and that all the...

Homework Statement If we have a transformation matrix \begin{bmatrix} 1 & 2 & 4 \\0 & 0 & 0 \\0 & 0 & 0 \end{bmatrix} Homework Equations The Attempt at a Solution I found the characteristic polynomial of this matrix: x^3 - x^2 = x^2(x-1) ...can anybody please help me...

I have a small idea on what irreducible and primitive polynomials are in Abstract algebra. But what is minimal polynomial? -Devanand T

Homework Statement Let A = \begin{pmatrix}1 & 1 & 0 & 0\\-1 & -1 & 0 & 0\\-2 & -2 & 2 & 1\\ 1 & 1 & -1 & 0 \end{pmatrix} The characteristic polynomial is f(x)=x^2(x-1)^2. Show that f(x) is also the minimal polynomial of A. Method 1: Find v having degree 4. Method 2: Find a vector v of...

Homework Statement Suppose that A is a 6x6 matrix with real values and has a min. poly of p(s) = s^3. a) Find the Characteristic polynomial of A b) What are the possibilities for the Jordan form of A? c) What are the possibilities of the rank of A? Homework Equations See below...

Homework Statement A n × n-matrix A satisfies the equation A2 = A. (a) List all possible characteristic polynomials of A. (b) Show that A is similar to a diagonal matrix Homework Equations The Attempt at a Solution A2 = A so, A2 - A = 0 A(A-I) = 0 Our minimal polynomial is x2 - x = m(x)...

For two matrices A and B, when p(A)=0 iff p(B)=0 for any polynomial, what will happen? i read that A and B have the same minimal polynomial, why?

1.Let F=K(u) where u is transcedental over the field K. If E is a field such that K contained in E contained in F, then Show that u is algebraic over E. Let a be any element of E that is not in K. Then a = f(u)/g(u) for some polynomials f(x), g(x) inK[x] 2.Let K contained in E...

1. Homework Statement [/b] Let V be the space of all polynomials of degree less than or equal to 2 over the reals. Define the transformation, H, as a mapping from V to R[x] by (Hp)(x)=\int^x_{-1}p(t)dt\\. a) Find the monic generator, d, which generates the ideal, M, containing the range of H...

Homework Statement Let V be the vector space of n x n matrices over the field F. Fix A \in V. Let T be the linear operator on V defined by T(B) = AB, for all B \in V. a). Show that the minimal polynomial for T equals the minimal polynomial for A. b) Find the matrix of T with respect...

About the invariance of similar linear operators and their minimal polynomial Notations: F denotes a field V denotes a vector space over F L(V) denotes a vector space whose members are linear operators from V to V itself and its field is F, then L(V) is an algebra where multiplication is...

I'm just learning a bit about the "minimal polynomial" today but there was a section from the book which I didn't understand. This is the section, and I've circled the bit I'm having trouble with. http://img15.imageshack.us/img15/1825/97503873.jpg (sorry, it won't let me post an image for...

Find the minimal polynomial with root 21/3 + 21/2. I would just use maple but I do not have it installed on this machine. I found the polynomial and verified that this is indeed a root. I only have Eisenstiens criterion for determining whether it is irreducible, and I can not apply it in...

Here is my problem: Let A be a complex n x n matrix with minimal polynomial q(x)=the sum from j=0 to m of \alpha_j x^j where m\leq n and \alpha_m = 1. Show: If A is non-singular then \alpha_0 does not equal 0. So, I get that 0=q(A)=\alpha_0 I_n + \alpha_1 A + \alpha_2 A^2 +...+A^m...

Homework Statement Let f(x) be an irreducible polynomial cubic in Q. For example f(x) = ax^3 + bx^2 + cx + d Let A be a 3 x 3 matrix with entries in Q such that char(A,x) = f(x). Find the minimal polynomial m(x) of A. Can you generalize to a degree n polynomial? Homework Equations...

Homework Statement Find the minimal polynomial of \frac{\sqrt{3}}{1+2^{1/3}} over Q we'll call this x Homework Equations I wish I knew some :( The Attempt at a SolutionBy taking powers of x, I was able to show that the extension Q(x) has degree six (since 21/3 and sqrt(3) are both...

Let A be an n x n matrix; denote its distinct eigenvalues by a_1,...,a_k and denote the index of a_i by d_i. How do I prove that the minimal polynomial is then: m_A(s) = (s-a_1)^d_1*...*(s-a_k)^d_k ? The characterstic polynomial is defined as: p_A(s) = (s-a_1)*...*(s-a_n);

f(t)=t^n+a_n-1t^(n-1)+...+a1t+a0 there's a square matrix of order n, A: \bordermatrix{ & & & & \cr 0 & 0 & ... & 0& -a_0 \cr 1 &0 & ... & 0 & -a_1 \cr ... & ... & ... & ... & ... \cr 0 & 0 & ... & 1 & -a_{n-1}\cr} show that f(t) is...

I have matrix A \left(\begin{array}{ccc}6&2&-2\\-2&2&2\\2&2&2\end{array} \right) Its characteristic polynomial is p(\lambda)=\lambda^3 - 10\lambda^2 + 32\lambda -32 Finding minimal polynomial i get...

Given a matrix A how can I found its minimal polynomial? I know how to find its characteristic polynomial, but how do I reduce it to minimal? Thanks, Chen