Tagged Questions
0
votes
1answer
25 views
Condition Number of Polynomial (Condition Number = 0)
I'm calculating the condition number of a polynomial equation
$$
y = (x-2)^{9}
$$
for this equation, the Jacobian is equal to ...
1
vote
1answer
36 views
Invariant subspaces and minimal polynomial
I wanted to know whether every invariant subspace $U$ of an endomorphism $A$ with minimal polynomial $m_A= \Pi_{i=1}^n p_i$, where the $p_i$ are mutually coprime polynomials, can be written in the ...
2
votes
2answers
30 views
When is the linear map of plugging $m$ numbers in a polynomial is surjective?
Let $n$, $m$ be positive integers and $V_n$ be the vector space of the polynomials of degree less than or equal to $n$ whose coefficients are complex numbers. For $m$ complex numbers $a_1,\dots,a_m$, ...
1
vote
1answer
50 views
Prove that the determinant of polynomials is zero
Prove that this determinant is zero (this matrix is $n\times n$):
$$\begin{vmatrix}
f_1(a_1) & f_1(a_2) & \cdots & f_1(a_n) \\
f_2(a_1) & f_2(a_2) & \cdots & f_2(a_n) \\
\vdots ...
1
vote
2answers
47 views
matrix representation of polynomial
Here is a polynomial $p(x,y) = (ax + by)^2$, it can be written like this $$p(x,y) = \left(\left[ \begin{array}{cc}
a & b \\
\end{array} \right] \left[ \begin{array}{c}
x\\
y\\
\end{array} ...
1
vote
0answers
55 views
Characteristic polynomial and a random polynomial in $\mathbb C$
Given $A \in M_{n \times n} (\mathbb C)$ with characteristic polynomial $P_A(x) = (x-\lambda_1)^{n_1} \cdot ... \cdot (x-\lambda_k)^{n_k}$. Let $g \in \mathbb {C [x]}$ a polynomial. Calculate the ...
1
vote
2answers
81 views
Find the characteristic polynomial of a matrix
I am trying to find the characteristic polynomial of: $$ A= \begin{pmatrix}
\alpha_1 & \alpha_2 & \cdots & \alpha_{55} & \\
\alpha_1 & \alpha_2 & \cdots & \alpha_{55} ...
1
vote
1answer
40 views
Quadratic Operator Notation?
I am dealing with functions that are linear combinations of:
$[x_1^2, x_2^2... x_n^2, x_1x_2, x_1x_3... x_n-1x_n]$
spanned over a column.
All these functions obey the law:
$F(aX) = a^2F(X)$ for ...
1
vote
1answer
90 views
Why are Vandermonde matrices invertible?
A Vandermonde-matrix is a matrix of this form:
$$\begin{pmatrix}
x_0^0 & \cdots & x_0^n \\
\vdots & \ddots & \vdots \\
x_n^0 & \cdots & x_n^n
\end{pmatrix} \in ...
1
vote
0answers
42 views
Image of the Sylvester matrix is the degree of the GCD
Let $P_k(F)$ denote the $F$-vector space of (univariate) polynomials of degree $\leq n$. Letting $F$ be a field lets everything be monic, but it seems sufficient to consider a ring $R$ such that the ...
1
vote
4answers
33 views
Prove that if $p(A)=0$ where A is a matrix of a linear operator ($A \in L(V)$), $p(\lambda)=0$ if $\lambda \in \sigma(A)$
I think it's all in the title. $p$ is some random polynomial.
I don't know how to approach this one. I've tried taking the roots of $p$, placing them on the diagonal of a new matrix and reasoning that ...
0
votes
0answers
39 views
Vandermonde matrix and polynomials
Question attached as image, deals with polynomials of order N and determinant of
Vanderbilt matrix.
0
votes
1answer
27 views
Equal Shape: Recovering an Isomorphism Between $M_{3\times 2}(F)$ and $P_5(F)$
I'm asked to find an isomorphism between $M_{3\times 2}(F)$ and $P_5(F)$, but what does it mean for a $3\times 2$ matrix to have an inverse?
1
vote
0answers
46 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
1answer
87 views
My proof that if for a k degree polynomial $P(x)$, for the matrix $A$, $P(A)=0$ then $A$ is invertible
Let $P(x)$ be a $k$-degree polynomial with with non-zero free coefficient. Prove that if for matrix $A$, $P(A)$=0, then $A$ is invertible and $A^{-1}$ is $k-1$ degree $A$ polynomial.
Here's my ...
