1
vote
2answers
33 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 ...
2
votes
0answers
32 views

Linear Independence of Powers of “roots vector” [duplicate]

Let us be working over the field of complex numbers. Suppose $f(x)= a_n x^n + \cdots +a_1 x + a_0$ is a degree $n$ polynomial with $n$ distinct roots $z_1,\ldots,z_n$. Is the following matrix always ...
0
votes
1answer
44 views

dimension of subspace - polynominals evaluated on f

I need to prove that the dimension of the subspace of endomorphisms is less or equal m, if m is the degree of a polynomial p of K[t] \ {0} with p(f) = 0 (f is endomorphism). In a second step I ...
3
votes
3answers
51 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
2
votes
1answer
46 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 $p$.
2
votes
1answer
41 views

Let P be the set of all polynomials of degree ≤ 3 such that p(t) = t. Is P a subspace of P3?

Let P be the set of all polynomials of degree ≤ 3 such that p(t) = t. Is P a subspace of P3? I'm not really sure how to solve this. I know that I have to prove that: Since p, q ∈ P3, k*p ∈ P3 and ...
1
vote
1answer
28 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
28 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\\ ...
0
votes
1answer
47 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 ...
1
vote
1answer
26 views

Polynomial in an endomorphism $u$ : justification?

In my course we defined the value of a polynomial $P=\sum_{k=0}^{n}\alpha_kX^k$ in the endomorphism $u$ of a vector space E: $$ P(u)=\sum_{k=0}^{n}\alpha_ku^k=\alpha_0Id_E+\alpha_1u+\cdots ...
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
75 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 ...
3
votes
0answers
48 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 ...
-1
votes
3answers
102 views

Does $s(0) = s(1)$ define a vector subspace in $\mathbb C[X]$?

I believe that $s(0) = s(1)$ does not define a vector space in $\mathbb C[X]$, but I am unsure how to show it. I know it doesn't satisfy the zero vector condition, nor is it closed under vector ...
3
votes
1answer
65 views

a polynomial about continuous function

Let $\{a_i(x):\mathbb{R}\rightarrow \mathbb{C}\}$ be continuous functions, does there exist some continuous functions $\{\lambda_i(x)\}$ such that $$a_{n-1}(x) y^n+a_{n-2}(x) y^{n-1}+\cdots ...
0
votes
1answer
29 views

If $abx^2 = (a-b)^2 (x+1)$ then $ [1 +(4/x)+(4/x^2)]^{(1/2) }=$?

As the title says. I found this question in our next term's book. A) (a+b)/(a-b) B) (a-b)/(a+b) C) a^2 +ab D) none
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 ...
0
votes
2answers
54 views

Determine whether this is a subspace of $P_3$

Let $S$ be the following subset of the vector space $P_3$ of all real polynomials $p$ of degree at most 3: $$S=\{p\in P_3\mid p(1)=0, p^\prime (1)=0\}$$ where $p^\prime$ is the derivative of $p$. ...
3
votes
1answer
21 views

Generic points as coefficients of polynomial kernels?

I am reading the paper Dual-to-Kernel Learning with Ideals. Here is part of it: The definition/motivation of genericity in Wikipedia are A generic point of the topological space $X$ is a point ...
1
vote
0answers
20 views

Show equivalence corresponding Nulls of function.

I'd like to show that the following two propositions are equivalent: (1) $f \in \mathbb{R}[x]$ has a multiple Null, so it's $\ge 2$. (2) $f$ and $f'$ have a common Null, whereas $f'$ describes the ...
0
votes
1answer
26 views

minimal polynomial and linear transformation

If $T:\Bbb{C} \to \Bbb{C}$ defined by $T(x)=x$ . T satisfity minimal poly is $x-1$. Is it correct. Any polynomial of degree $>1$ is a linear transformation on C .this type of transformation exist ...
2
votes
0answers
16 views

Give bounds for degree of “decreasing” polynomial

Let $p$ be a polynomial of minimal degree to which the following is true: $p(0) > \sum\limits_{i=1}^n \lvert p(i) \rvert + \sum\limits_{i=1}^m \lvert p(-i) \rvert$ Give upper and lower ...
0
votes
3answers
38 views

Existence of a unique polynomial

Suppose $z_1, .....z_{m+1}$ are distinct elements of $F$ and $w_1,....,w_{m+1} \in F$. Prove that there exists a unique polynomial $p \in P_m(F)$ such that $p(z_j) = w_j$ for j=1,...m+1. Any ideas on ...
0
votes
2answers
29 views

Resultant$(f,g)$ says when there exist $\phi,\psi$ such that $\psi f + \phi g = 0$. How do I actually find them?

If $f$ and $g \in k[X]$ are two polynomials such that $\textrm{Res }(f,g)=0$ how do I find $\phi$ and $\psi$ with $\deg \phi < \deg f$ and $\deg \psi < \deg g$ such that $$\psi f +\phi g =0$$
1
vote
1answer
32 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 ...
0
votes
1answer
56 views

Perturbation theory for algebraic equations

I'm trying to find expansion (up to the 2nd non zero term) for the roots of: $x^5-x^2+\epsilon=0$ as $\epsilon\rightarrow0$ So I've assumed the solution may be written as a power series ...
1
vote
0answers
32 views

Diagonalization of a linear transformation in the polynomial vector space

Let $V = R_3[X]$ be the vector space of polynomials with real coefficients of degree at most 3 and consider the linear transformation $V \rightarrow V$ defined by $f_a(p(x))=p(1-ax)$ for each $p(x) ...
1
vote
1answer
48 views

Dimension of the vector space of homogeneous polynomials

Let $k[X_0, X_1, \ldots, X_n]_d$, or briefly $k[X]_d$, be the $k$-vector space whose elements are the zero polynomial and homogeneous polynomials of degree $d\geq 1$. I found the following formula for ...
5
votes
2answers
234 views

Is evaluation homomorphism surjective?

Let $A^n$ be an affine space over $\mathbb{C}$ and let $\mathbb{C}[X_1,\cdots,X_n]$ be the polynomial ring of $n$ variables. Then $A^n\to (\mathbb{C}[X_1,\cdots,X_n])^*$ by evaluation homomorphism, ...
0
votes
1answer
35 views

How to simplify floor polynomial given lower bound on x?

$$ \left\lfloor\frac{8x^2 + 5x -4}{3x^2 + x}\right\rfloor $$ where $x$ > $\sqrt{8}$ How would you simplify this type of expression? *Please note the floor operation surrounding the expression ...
1
vote
2answers
32 views

Show inequality for two elements in $\mathbb{R}^n$

I know that $x,y\in \mathbb{R}^n$ are such that $x_1\leq0,x_1^2\geq x_2^2+\dots+x_n^2$ and $y_1\geq 0,y_1^2\geq y_2^2+\dots+y_n^2$. Is it possible to show that $$x_1y_1+x_2y_2+\dots +x_ny_n\leq 0$$ ...
1
vote
2answers
59 views

dimension of vector space & polynomial

Say that a polynomial with real coefficients in two variable, $ x, y $ , is balanced if the average value of the polynomial on each circle centred at the origin is $ 0 $ . The balanced polynomials of ...
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 ...
2
votes
1answer
64 views

Field not closed under complex conjugation

What is an example of an algebraic field not closed under complex conjugation? In all subfields of $\mathbb C$ I think of, complex conjugation is a transposition. I think I understand that it is ...
0
votes
1answer
38 views

Simple Math Question

How can we find the answer for this by solving without actually calculating.... $$\sqrt{\frac{(0.75)^3} {1 - (0.75) }+{(0.75 + (0.75)^2 + 1)}}$$ Actually I meant for the first square-root sign ...
0
votes
3answers
36 views

Simple Trigonometry and algebra

If $$\sec\theta = X + \frac{1}{4X},$$ then what is $${\sec\theta + \tan\theta}$$ in terms of $X$?
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
35 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
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
22 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
3answers
49 views

Is there a polynomial $f\in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$

I was asked the following question: $g\in \mathbb Q[x]$ is a polynomial (not the zero polynomial). Find $f \in \mathbb Q[x]$ such that $f(x)^2=g(x)^2(x^2+1)$ or show that such an $f$ does not exist. ...
0
votes
2answers
60 views

Simple yet confusing: if $ f^2(x)=g^2(x)(x^2+1) $ then $gcd( f^2(x),g^2(x))=(x^2+1)$?

As mentioned in the title: f(x) and g(x) are polynomials above the Rationals field. if $ f^2(x)=g^2(x)(x^2+1) $ then does it mean that $ gcd( f^2(x),g^2(x))=(x^2+1) $? or maybe it isn't the ...
2
votes
1answer
177 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 ...
4
votes
2answers
178 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
2answers
64 views

Orthogonal polynomials on $[0,1]$

Are the orthogonal polynomials for the standard $L^2$ product on $[0,1]$ well-known? I couldn't find anything upon a quick web search.
1
vote
1answer
37 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 ...