0
votes
0answers
22 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
47 views

Inverse of a matrix is expressable 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
31 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
32 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
29 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
27 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
79 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
25 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
35 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
42 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
66 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
62 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
62 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
37 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
50 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
50 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
23 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
184 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
33 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
14 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
1answer
39 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
55 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
40 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
28 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
42 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
162 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
120 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
59 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
62 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
66 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 ...
2
votes
0answers
54 views

rank of formal derivation

Let $K$ be a field, $n \in \mathbb{N}_{>0}$ and $K[x]_{ \leq n} $ he space of polynomials above $K$, that have a maximum degree of $n$. We define the formal derivation as follow: $\frac{d}{dx}= ...
6
votes
0answers
49 views

How many pairs of nilpotent, commuting matrices are there in $M_n(\mathbb{F}_q)$?

As a follow-up to this question, I've been doing some work counting pairs of commuting, nilpotent, $n\times n$ matrices over $\mathbb{F}_q$. So far, I believe that for $n=2$, there are $q^3+q^2-q$ ...
2
votes
1answer
69 views

Linear Algebra: Finding the matrix representation with respect to standard basis

I would appreciate some help with a linear algebra practice question, I'm studying for my final and I am stuck, this is a screenshot of the question: Are my answers correct? a) $P_{2}$: $ ...
1
vote
0answers
90 views

Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ ...
2
votes
1answer
58 views

Sylvester matrix and GCD degree

How to prove that the degree of a $\gcd$ of two polynomials is equal to the dimension of the null space of the Sylvester matrix? I know that any linear combination of the rows of $S(u,v)$ is a linear ...
2
votes
1answer
101 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
30 views

Question about Jordan form - Linear algebra

Quick question, We are given that the characteristic polynomial of a matrix $A$ is $P_A(x)=(x-1)^4$ We are asked to find all the possible jordan forms of $A$. Obviously the minimal polynomial of ...
2
votes
1answer
50 views

Find minimal polynomial of a difficult transformation

we are asked to find the minimal polynomial of the transformation: $T: M_n(\mathbb C) -> M_n(\mathbb C)$ $T(A)=CA$ when $C$ is a diagonal matrix with the values $c_1,c_2,c_3,...,c_n$ on the ...
0
votes
0answers
21 views

Find the minimal polynomial using other matrix's minimal polynomial

A is nxn matrix and MA is it's minimal polynomial. Write M(I-A) (the minimal polynomial of I-A) by using MA.(express it by MA) Thoughts (that's haven't lead me anywhere by now): We know that MA(A)=0 ...
0
votes
1answer
42 views

Find minimal polynomial of this transformation

We are asked to find the minimal polynomial of this transformation: $T:\mathbb C_n[x] -> \mathbb C_n[x]$ $T(p) = p'+p$ What I did: I found the transformation matrix with respect to the standard ...
1
vote
2answers
57 views

Show that the matrix is invertible

let $A \in M_n(F)$ be a n by n matrix with values from an unknown field $F$. $P_A(t)$ is the characteristic polynomial of $A$, and $g(t) \in F[t]$ a polynomial of an unknown degree. assume that ...
1
vote
2answers
63 views

Why doesn't the minimal polynomial of a matrix change if we extend the field?

Why doesn't the minimal polynomial of a matrix change if we extend the field? I appreciate any help or proof.
0
votes
1answer
72 views

Find the Characteristic polynomial

The characteristic polynomial of $A \in M_{4}(\Bbb R)$ is: $P(t)= t^4-t$ Find the Characteristic polynomial of: $A^2, A^4$ ($A^4$ was easy but with $A^2$ I'm stuck) Same question with the field ...
1
vote
2answers
30 views

What can we say about $\dim \operatorname{null}(AB)$ from knowing $p_A$ and $p_B$?

Say, there are two matrices $A, B \in \mathbb R^{3,3} $ such that their characteristic polynomials are $p_A(t) = t^3 − t^2 + 2t$ and $p_B(t) = t^3 − 7t^2 + 9t − 3$. What do we know about $\dim ...
1
vote
1answer
30 views

homework - Show a matrix as a combination of other matrices and long division

The topic we are dealing with here is polynomial division. The question is: We are given a polynomial: $f(x) = (x+1)(x-1)^2$, and a matrix $D \in R^{nxn}$ such that $f(D)=0$ Using only $I, D, D^2$ ...
2
votes
2answers
55 views

Given $A, B\in R^{n\times n}$ diagonal matrices, there exist $p,q \in R[x]$ and $X\in R^{n\times n}$ such that $A = p(X),B=q(X)$

(1) We are given $A,B \in R^{n\times n}$ diagonal matrices of n rows and n columns with real values. Show that there are $X \in R^{n\times n}$ and polynomials $q$ and $p$ such that: ...
1
vote
0answers
51 views

homework: rings, matrices and polynomials

A,B are both nxn and diagonal matrices. Prove that there is a matrix X which is nxn, and polynomials p and q such that A= p(X), B= q(X) Is this true for ANY 2 matrices (we do not assume that they're ...
0
votes
1answer
187 views

Find a basis for ker(T) and range(T) for the given transformation and compute T(5x-4)

I am not really having trouble with $a)$, $c)$, $d)$ or $e)$. For $a)$ I put that $B$ is a basis for $\mathbb{P}^1$ because it has ${\rm dim} = 2$ and the highest degree is 1, and for $B'$ it has ...
1
vote
1answer
50 views

Matrix polynomial factorization

This is about exercise 1207 from the book "Problems and Solutions in Mathematics", 2nd edition, by Ta-Tsien. Let $p$ be a prime and let $V$ be an $n$-dimensional vector space over the finite field ...