0
votes
0answers
8 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 $w_i \neq z_j$ ...
6
votes
4answers
377 views

Determinant of a matrix with $t$ in all off-diagonal entries.

It seems from playing around with small values of $n$ that $$ \det \left( \begin{array}{ccccc} -1 & t & t & \dots & t\\ t & -1 & t & \dots & t\\ t & t & -1 ...
-1
votes
2answers
37 views

How to get A,B and C given XYZ?

How do I get $a$, $b$, and $c$? Given $$X=\frac{a+\frac{1}2b}{a+b+c}$$ $$Y=\frac{b(\frac{\sqrt3}{2})}{a+b+c}$$ $$Z=\frac{a+b+c}{3}$$ in other words How do i get $a$, $b$, and $c$ on the left ...
2
votes
2answers
69 views

“Conic sections” that are really just two straight lines

My teacher was teaching co-ordinate geometry and today he said that the following equation will always represent a conic section:$$ax^2+by^2+2hxy+2gx+2fy+c=0$$ Then he said that if the determinant of ...
1
vote
3answers
71 views

Does substitute $\lambda$ with matrix $A$ in a polynomial conflict with the Axiom of Substitution?

This seems to be an elementary question, gonna ask it anyway. Suppose that $A$ is a square matrix, and that $p(x)$ is its characteristic polynomial, we know that (1) $p(x) = \det(xE - A)$ We also ...
0
votes
0answers
52 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
19 views

Transformation matrix of a polynomial

I would really appretiate some help about the following transformation matrices. We have to write a tranformation matrix in basis $B = \{ 1 + x, x + x^2, x^2 \}$ with a polynomial $(Ap)(x) = (x^2 - ...
0
votes
1answer
33 views

Linear Equation as matrix

Using a series of 3x3 matrices multiplied together, it is possible to create a matrix which will rotate, translate, scale and invert a size 2 vector. Using a 4x4, it is possible to do this to a size ...
1
vote
1answer
47 views

Characteristic polynomial $p_{cA}(t)$

Let's define $p_{A}(t)$ the characteristic polynomial of square matrix $A$ over $R$. Prove that for every $c \in R$, $c \ne 0$ the characteristic polynomial $p_{A}(t)$ of the matrix $cA$ is ...
0
votes
4answers
80 views

Prove that the characteristic polynomial of a nilpotent matrix is $x^n$

How can I prove that the char.pol. of a nilpotent matrix is of the form $x^k$? I'm trying to do it by contradiction but assuming that $p_{xA}=a_0+a_1x+\dots+a_mx^m+\dots+a_nx^n$ seems not giving any ...
0
votes
1answer
42 views

prove that if T is invertible transformation there is polynomial $p$ such that $T^{-1} = p(T) $

I know how to prove this using Hamilton.C but something doesn't make sense to me. if I assume that there is such polynomial p(x), so p(T)T = I . then looking at these polynomials I get: p(x)x = 1 so ...
0
votes
1answer
20 views

Question about calculating exponent of polynomial

$V=R_{3}[X] $ and $T:V->V$ is a linear transformation : $T(p(x)) = p(x) + xp'(x)$ I need to find $e^{T(1+x+x^{2}-x^{3})}$ I don't understand how to do it? what does it mean to calculate exponent ...
0
votes
0answers
25 views

Finding matrix index from triangular array offset

I have a mapping from a lower triangular matrix, A, to a vector,v: A(i,j) -> v( $\lfloor i(i+1)/2 \rfloor + j$ ) $i,j\in[0,N]$, $j\leq i$, $N\in\cal{N}$, $N\geq 0$ (so, my first row is row 0, and ...
1
vote
0answers
21 views

Linear Transformation - linear algebra question [duplicate]

$T:\mathbb{R}_2[x] \mapsto \mathbb{R}_2[x]$ s.t.: $$ \begin{array}{l} T(1) = 3+2x+4x^2, \\ T(x) = 2+2x^2, \\ T(x^2) = 4+2x+3x^2. \end{array} $$ Is there base $B$ of $\mathbb{R}_2[x]$ that $[T]_B = ...
0
votes
1answer
24 views

find minimal polynomial of $T(p)=p'+p$

I'm trying to solve the following question: let $T: \mathbb C_n[x] \to \mathbb C_n[x]$, $T(p)=p'+p$ find the characteristic and minimal polynomial of $T$. What I'm trying to do is the following: I ...
0
votes
4answers
20 views

What is the matrix corresponding it a linear transformation of a polynomial?

Given the linear map $T(f(x)) = f(2x+1)$ where $f(x)$ is a polynomial of degree $3$, what is the matrix corresponding to $T$?
2
votes
3answers
30 views

Finding the characteristic polynomial in a square matrix

The example in the textbook had a square matrix \begin{pmatrix} 0&1&0\\0&0&1\\4&-17&8 \end{pmatrix} Then proceed to say $ \ (\lambda \cdot I - A) \ $ is \begin{pmatrix} ...
1
vote
0answers
48 views

Looking for ideas on the properties of a simple matrix

I am interested in investigating the "theory" of this simple matrix sequence which I ran across in my studies. $$A_n(x)=\begin{pmatrix} 1 &x \\ -x& 1 \end{pmatrix}^n$$ E.g. ...
2
votes
1answer
105 views

Polynomial of matrix

The question here is that, is it possible to solve a polynomial of matrix like the following $A^{2}+A=B$, where $B$ is a known semi-definite matrix, and $A$ is the unknown symmetric matrix we ...
0
votes
0answers
31 views

PSD matrix and non-negative polynomial

So I'm trying to prove that if there exists a $5 \times 5$ matrix $Q$ such that $$Q \succeq0,\,\, a_{l-1} = \sum\limits_{i+j=l} Q_{ij} , l=1,\ldots,5$$ then there exists a fourth degree polynomial ...
2
votes
1answer
107 views

Inverse of a matrix is expressible as a polynomial?

Let $A$ be an $n \times n$ matrix. Prove that if A is invertible, then there exists a polynomial $p$, such that $A^{-1}=p(A)$ Thus far: Let $W$ denote the $k$ dimensional A-cyclic subspace spanned ...
0
votes
2answers
32 views

Proving that an eigenvalue is a root of a polynomial

Let $A$ be an $n \times n$ matrix, and let $\lambda$ be an eigenvalue of A. Prove that if $p$ is a polynomial such that $p(A)=\mathbb{0}$ then $\lambda$ is a root of $p$.
1
vote
1answer
41 views

Transformation Matrix $M_B^B$ of $P_3$ for $B = (1,x,x^2,x^3)$. Is that correct?

I have the following task and just wanted to check weather this is (written) correct(ly). Let $V$ be the vector space of all polynomials of grade $\le 3$ and $f: V \rightarrow V, p \rightarrow p'$ an ...
0
votes
1answer
39 views

Change Bases of Linear Transformation

I have: T: $P_2(R)\to P_1(R)$ $T(a + bx + cx^2) = (a - 3b + c) + (2a - 6b + 3c)x$ Need to find bases $\alpha' ,$ $\beta'$ such that $[T]_{\alpha'\beta'}$ is reduced echelon form of ...
1
vote
0answers
46 views

Calculate determinant of Vandermonde using specified steps.

$V_n(a_1,a_2\dots, a_n)$ is a $N\times N$ Vandermonde matrix = $$\left|\begin{array}[cccc] 11&z_1&\cdots&z^{n-1}_1\\ 1&z_2&\cdots&z^{n-1}_2\\ ...
1
vote
1answer
18 views

prove that if gcd(f, minimal polyonimial of A) is not 1 then f(A) is not invertible

A is square matrix and f is polynomial. prove that if gcd(f, minimal polyonimial of A) is not 1 then f(A) is not invertible. any hints please..
3
votes
2answers
114 views

Tricky Question on Induction and Characteristic Polynomials

I am to prove via induction that for any $n \times n$ matrix $A$, the characteristic polynomial of $A$ has degree $n$; $(-1)^n$ as the coefficient of the $\lambda ^n$ terms; $(-1)^{n-1}\cdot ...
1
vote
1answer
27 views

Linear algebra, question about polynoms

A,B are matrices n*n over a field F. I am given a polynom f(t) {belongs to F[t]} . How can I show that Af(BA)B= ABf(AB)? I defined a polynom g(t)= t*f(t). Then I substituted AB instead of t, but I ...
1
vote
1answer
56 views

Minimal and Characteristic Polynomials of Matrix Multiplication Transformation

Fix a matrix $A \in M_n(F)$ where $F$ is a field, and consider the following linear transformation $\phi_A: M_n(F) \to M_n(F)$ given by $\phi(B) = AB$. Prove that the minimal polynomials of $\phi$ and ...
1
vote
2answers
50 views

matrix and polynomial

Let $a,b,c$ be all the roots of $ x^3 + sx + t$. What is the determinant of the matrix $\begin{bmatrix} a & b & c\\ b & c & a\\ c & a & b \end{bmatrix}$? I wrote ...
0
votes
2answers
69 views

Is the space isomorphic?

$\mathcal{P}_5$ and $\mathbb{R}^5$. So $\mathbb{R}^5$ has a dimension of 5, but how do you determine the dimensions of $\mathcal{P}_5$? Any element of $\mathcal{P}_5$ is of the form ...
1
vote
2answers
67 views

$A^m = r_m(A)?$ Power of a matrix!

In my Linear Algebra textbook we are reading, the following is stated for computing the power of a matrix in one of the advanced chapters as an exercise, $A^m = r_m(A)$. $r_m (A)$ is the remainder ...
1
vote
1answer
63 views

Find $f\left(A\right)$ for a polynomial function of a square matrix

So here is the complete question: Use the given definition to find $f\left(A\right)$: if $f$ is the polynomial function $f\left(x\right)= a_0+a_1x+a_2x^2+...+a_nx^n$ then for a square matrix ...
1
vote
1answer
42 views

Is it true that the constant in the characteristic polynomial is $(-1)^n det A$?

A is nxn matrix with the characteristic polynomial Pa(t). Is it true that the constant in the characteristic polynomial is $(-1)^n det A$? Please help me, I have a test tomorrow.Thanks for the help.
2
votes
2answers
53 views

Factor 9 terms with 3 variables into 4 expression

I just got the determinant from a 4x4 matrix and the simplified version is below. $$ det(M) = \begin{vmatrix} 2k-mw^2 & -k & 0 & 0 \\ -k & 2k-mw^2 & -k & 0 \\ 0 & -k ...
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, ...
1
vote
1answer
29 views

A is Mn×n(C) with rank r and m(t) is the minimal polynomial of A. Prove deg $m(t) \leq r+1$

$A$ is a matrix of $M_{n \times n}(\mathbb{C})$ with rank $r$ and $m(t)$ is the minimal polynomial of A. I need to prove that : deg $m(t) \leq r+1$ I need to find a condition of the matrix A, in ...
2
votes
1answer
249 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 ...
0
votes
1answer
34 views

Dividing two polynomials as vectors

I'm trying to write a program that divides two polynomials in R1. Something tells me this can be done with matrices but I'm not sure what the algorithm for this is. If I represent the two polynomials ...
0
votes
0answers
16 views

Univariate and Linear Representation Lemma

I'm trying to understand the proof of the lemma: Let $\mathbb{F}$ be a finite field with $q$ elements and $\mathbb{K}$ an extension field of degree $n$ of $\mathbb{F}$. Let $A$ be a linear mapping ...
1
vote
2answers
60 views

Linear Algebra - Given the Jordan form of $A \in Mat_7(\mathbb F)$, find Jordan form of $A^2+A+I_7$

Given that the jordan form of the matrix $A \in Mat_7(\mathbb F)$ is: $\begin{pmatrix} J_2(1) &\cdots &0\\0& \cdots J_3(1) \cdots &0\\0&0& \cdots J_2(2)\end{pmatrix}$ Find ...
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 ...
1
vote
1answer
44 views

How to find charpoly from eigenvalues and CH to prove an equation

For an uknown 3x3 matrix $A$ we know that $\operatorname{tr} A = 0$, $\det(A) = 1/4$ and we also know that two eigenvalues are the same. Proove that $4A^3 = -3A - I$. Problem says to use Vieta to find ...
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
1answer
57 views

Matrix for linear map involving polynomials

I need to find the matrix corresponding to the linear map $f:V_3 \rightarrow V_3$, where $V_3$ is the vector space of all polynomials of degree less than or equal to 3, $$f(p(X))=p(X)-p'(X)$$, with ...
4
votes
2answers
178 views

If $(A-\lambda{I})$ is $\lambda$-similar to $(B-\lambda{I})$ then $A$ is similar to $B$

When reading the topic about primary and rational canonical form of matrices I stuck myself on this theorem: The matrices $A,B\in K^{n\times n}$ are similar if and only if their characteristic ...
2
votes
1answer
163 views

Finding characteristic polynomial of adjacency matrix

Short question im having a tad difficulty with. I'm trying to find the characteristic polynomial of a graph that is just a circle with n vertices and n edges. I think the adjacency matrix should ...
2
votes
1answer
64 views

Nondiagonalizable Matrix and Polynomials

I got the following problem: If $A$ is a nondiagonalizable square matrix of order $n$ over field $\mathbb{F}$ then there exists a polynomial $P$ of degree $n-1$ over $\mathbb{F}$ such that ...
0
votes
3answers
68 views

About Jordan Form

For a $A\in M_n({\bf C})$ with a minimal polynomial $m(x) = (x-c)^n$ then we have a Jordan form wrt some basis $$ A=\left( \begin{array}{ccccc} ...
3
votes
1answer
75 views

Show that T is a linear transformation and find a, b, c

I'm having trouble understanding this question and the proper way to solve it. I don't understand the solution given and why this was the right way to answer it. Problem: For the vector space ...