0
votes
0answers
7 views

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 $w_i \neq z_j$ ...
0
votes
0answers
50 views

Eigenvalue formula for 4x4 symmetric matrix

Is there a formula/algorithm that is accurate to used in finite precision arithmetic (aka numerical stable ) for small symmetric matrix of size 4x4. Additionally I'm looking if it require similar ...
0
votes
2answers
23 views

Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
2
votes
0answers
68 views

Criteria for a solvable septic equation

I have a certain 7x7 matrix whose elements are all symbolic, and I want to know the eigenvalues. I have to solve a septic equation, but it is generally impossible. However, I am only interested in the ...
0
votes
2answers
70 views

Calculating the characteristic polynomial

I'm stuck with this problem, so I've got the following matrix: $$A = \begin{bmatrix} 4& 6 & 10\\ 3& 10 & 13\\ -2&-6 &-8 \end{bmatrix}$$ Which gives me the following ...
2
votes
1answer
83 views

Proof that a is an eigen value of p(T) if and only if a=p(lambda) for some eigenvalue lambda of T

$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\F}{\mathbb{F}} \newcommand{\C}{\mathbb{C}} \newcommand{\LM}{\mathcal{L}}$ Question: Suppose $\F = \C, T \in \LM(V), p \in ...
-3
votes
1answer
50 views

Finding an eigenbasis of an operator.

Consider the operator $T:\mathbf{P}_1\rightarrow\mathbf{P}_1$ by $T(ax+b):=-bx+(2a-3b)$. I want to find out if there is a basis of eigenvectors of this operator $T$. Now, I have no idea what to do, ...
1
vote
1answer
76 views

Is my proof correct? linear transformation over $\Bbb R$ has invariant subspace $\dim(U)=1$ or $2$

$V$ is a Vector space over $\Bbb R$, and $\dim(V)=n$. A linear transformation $T$ from $V$ to $V$. Then, T has an invariant subspace $U$ such that $\dim(U)=1$ or $2$ I read in many books, which ...
1
vote
3answers
22 views

Let $f: V_3 \rightarrow V_3$ be the function such that $p(X) \mapsto p''(X)$, calculate the eigenvalues of f

Let $V_3$ be the vector space of all polynomials of degree less than or equal to 3. The linear map $f: V_3 \rightarrow V_3$ is given by $p(X) \mapsto p''(X)$. Calculate the eigenvalues of f. First of ...
1
vote
2answers
56 views

Finding real, distinct eigenvalues for arbitrary constants

Let $A= \begin{bmatrix} 1 & 1 & 0 \\ -4 & -3 & 1 \\ k & 0 & 0 \end{bmatrix}$. Find all values of $k$ such that $A$ has three real distinct eigenvalues. I have obtained the ...
0
votes
1answer
57 views

Why is $ det(A - \lambda I) = (-1)^n \cdot [\lambda^n + c_1\lambda^{n-1} + … + c_n ] $?

Well the title tells you everything I want to know. Why is $ \det(A - \lambda I) = (-1)^n \cdot [\lambda^n + c_1\lambda^{n-1} + ... + c_n ] $ ? With this I then want to show that $ \det(A - \lambda ...
3
votes
0answers
73 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
2
votes
0answers
62 views

The minimal polynomial of A is dividing $x^{2013} -1$, prove A is diagonalizable over the complex field

$A $ is $nxn$ real matrix. The minimal polynomial of A is dividing $x^{2013} -1$. I need to prove that: (1). A is diagonalizable over the complex field. (2). If A is diagonalizable over the reals, ...
2
votes
1answer
248 views

Eigenvalues of 3x3 Covariance Matrix, Geometric Interpretation

Problem Definition I would like to code an algorithm for decomposing a covariance matrix into its eigensolution (set of eigenvalues and corresponding eigenvectors. In my specific case I want to deal ...
4
votes
2answers
189 views

Linear algebra : eigenvalues of an integral operator on polynomials

Consider the linear transformation $$ T : \left\{ \begin{array}{ccc} \mathbb{R}_n[X] & \to & \mathbb{R}_n[X] \\ P & \mapsto & \int_0^1 (X + t)^n\,P(t)\,dt \end{array}\right. $$ where ...
0
votes
3answers
72 views

Eigenvalue and Eigenvector proof of matrix polynomial

Consider a polynomial $$f(s) = s^m+a_1s^{m-1}+\cdots +a_m$$ and a square matrix $A$. Prove: if $\lambda$ is an eigenvalue of $A$ with corresponding eigenvector $x$, then $f(\lambda)$ is an eigenvalue ...
3
votes
4answers
191 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
1
vote
1answer
40 views

How to verify if characteristic equation is right?

I am new to EigenValues and EigenVectors. I am trying to solve a basic sum and somehow I am going wrong. The formula I know to get the characteristic equation is: $\lambda^3 - \sum(\text{diagonal ...
2
votes
2answers
94 views

Find eigenvectors of an infinite dimensional vectorspace Pn(R)?

Define a linear operator T:Pn(R)--->Pn(R) by T(p(x))=p(x)+p(2)x. (a) How many distinct eigenvalues does T have? (b) What are the dimensions of their corresponding eigenspaces? At first I started ...
2
votes
1answer
184 views

Relation between Algebraic multiplicities and rank of a matrix

A is a 6x6 matrix, $rank(A-3I) = 4$, the minimal polynomial of A is $(x-1)^2(x-3)^2$ I need to write the Jordan matrix options for A. How can I use the given information about the rank, what does ...
1
vote
1answer
54 views

Eigenvalue of $A$ is root of $p(t)$

I'm working on this proof problem: I have part (a) done. I think this is the Cayley Hamilton Theorem, so I got some insight from Wikipedia. I am a little confused with part (b) though because I ...
3
votes
1answer
663 views

How to prove “eigenvalues of polynomial of matrix $A$ = polynomial of eigenvalues of matrix $A$ ”

Title looks a little bit twisted. What I want to say is the following: $A\in\mathbb{R}^{n\times n}$, polynomial of matrix $A$: P(A)=$\displaystyle \sum_{k=0}^{n} c_k A^k$. $\lambda(A)$ is the set of ...
3
votes
1answer
109 views

What does $x_n = s\, x_{n-1}$ mean in the components of recurrence?

Say you have a reucrrence $x_{n+1} = 3x_n+2$. Though, it is a inhomogenous, it can be represented by a linear system $$\begin{bmatrix} x_{n+1}\\1\end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 0 & ...
1
vote
2answers
100 views

Find the characteristic polynomial of a matrix

I am trying to find the characteristic polynomial of: $$ A= \begin{pmatrix} \alpha_1 & \alpha_2 & \cdots & \alpha_{55} & \\ \alpha_1 & \alpha_2 & \cdots & \alpha_{55} ...
2
votes
1answer
68 views

Find Eigenvectors of a homomorphism over polynomial vector spaces

Let $\mathbb{R}[x]$ be the real-valued vector space of polynomials with real-valued coefficients and $F: \mathbb{R}[x]\rightarrow\mathbb{R}[x]$ be a homomorphism defined as $$ ...
1
vote
0answers
49 views

Linear Algebra: Linear transformation and eigenvalues [duplicate]

Hi could some one please help. I am having problems proving this. Let $A$ be an $n \times n$ matrix with complex entries and let $f (t) =\det(A - tI)$ be its characteristic polynomial. Prove ...
6
votes
2answers
2k views

Find the roots of a polynomial using its companion matrix

I would like to find the roots of a polynomial using its companion matrix. The polynomial is ${p(x) = x^4-10x^2+9}$ The companion matrix $M$ is $M={\left[ \begin{array}{cccc} 0 & 0 & 0 ...
5
votes
2answers
4k views

Minimal polynomials and characteristic polynomials

I am trying to understand the similarities and differences between the minimal polynomial and characteristic polynomial of Matrices. When are the minimal polynomial and characteristic polynomial the ...
1
vote
1answer
112 views

minimal polynomial of normal endomorphism with given eigenvalues

What's the minimal polynomial of a normal endomorphism $\phi$ with eigenvalues $2, 2, 1+i, 1+i, 1-i, 1-i, 3$? It is $\mu_\phi | (t-2)^2(t-1-i)^2(t-1+i)^2(t-3)$ but is there any more I can derive from ...
4
votes
1answer
925 views

Quick ways to _verify_ determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors …

What are easy and quick ways to verify determinant, minimal polynomial, characteristic polynomial, eigenvalues, eigenvectors after calculating them? So if I calculated determinant, minimal ...
3
votes
5answers
392 views

How to compute the characteristic polynomial of $A$

The matrix associated with $f$ is: $$ \left(\begin{array}{rrr} 3 & -1 & -1 \\ -1 & 3 & -1 \\ -1 & -1 & 3 \end{array}\right) . $$ First, I am going to find ...
3
votes
1answer
330 views

How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?

A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post How to calculate complex ...
2
votes
2answers
143 views

Computation of characteristic polynomial fails for me

For a matrix $A\in\mathbb{K}^{n\times n}$ where $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ the characteristic polynomial is defined as $$\chi_A(\lambda) := \text{det}(A-\lambda I_n) = \sum_{k=0}^n c_k ...