Tagged Questions

1
vote
1answer
31 views

If $f(X) = a_0 + a_1 X + a_2 X^2 \in \mathbb{F}[X]$ then show $f$ is uniquely determined by $f(x)$, $f(y)$, $f(z)$?

This is the exact question: It's part(ii) that I don't understand - what does it mean and what is it asking me to do? How would I go about constructing a proof? Any help would be much appreciated.
13
votes
2answers
146 views

Is a linear combination of minors irreducible?

Let $X=(X_{ij})_{1\le i,j\le n}$ be a matrix of indeterminates over $\mathbb C$. For choices $I,J\subseteq\{1,\ldots,n\}$ with $|I|=|J|=k$ denote by $X_{I\times J}$ the matrix $(X_{ij})_{i\in I,j\in ...
3
votes
1answer
49 views

Why Vandermonde's determinant divides such determinant?

Assume that $$ W(x_1,...,x_n;k)=\left [ \begin{array}{rrrrrrrr} 1 & x_1 &... & x_1^{n-2} & x_1^k \\ 1 & x_2 &... & x_2^{n-2} & x_k \\ & & \ddots \\ 1 & ...
1
vote
0answers
32 views

Why the ith coefficient of $|\lambda I-A|$ is the sum of all $i$-th order principle minors of $A$?

I come across a theorem that $f(\lambda )=|\lambda I-A|$, which equals to $\lambda ^{n}-a_{1}\lambda ^{n-1}+\alpha _{2}\lambda ^{n-2}-...(-1)^{n}a_{n}$ where $a_{i}$ is the sums of all ith order ...
2
votes
2answers
37 views

why this is correct: $\det(C+Di)$ is not zero, then there exists some real number $a$ such that $\det(C + a D)$ is not zero

I wonder why the following statement is correct: supposing $C$ and $D$ are two real matrix, if the determinant of the complex matrix $C + D i $ is not zero, then there exists some real number $a$ ...
-1
votes
1answer
37 views

divisibility of polynomials and determinant relations

Let $A$ be an integral domain and $f(x), g(x) \in A[x_1,\cdots,x_n]$. Write $f(x)=\sum \alpha_{\omega} x^{\omega}, g(x) = \sum \beta_{\omega} x^{\omega}$ where $\omega = (\omega_1,\cdots,\omega_n)$ ...
2
votes
3answers
104 views

Correlation between polynomial equations and matrix determinants

Expanding $p(x)=(ax-b)(cx+d)$ we get $acx^2+(ad-bc)x-bd$. Notice the determinant of the matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix} $ is $ad-bc$ exactly like the constant of $x$ ...
5
votes
4answers
210 views

Vector space of polynomials over $\mathbb{R}$ with degree $\leqslant n-1$

Let $P \in \mathbb{R}_{n-1}[X]$ be a polynomial of degree $n-1 \geqslant 0$. Let $\mathbb{R}_{n-1}[X]$ be the vector space of polynomials with degree $\leqslant n-1$ over $\mathbb{R}$. Show ...
1
vote
1answer
92 views

expressing product as Vandermonde determinants

Is it possible to express the product: $$ \frac{\prod_{i < j} (a_i - a_j)(b_i - b_j) }{\prod_{i,j} (a_i - b_j) }$$ as the determinant of a single matrix ? This comes from a physics paper. Should ...
2
votes
1answer
70 views

The derivative of characterestic polynomial?

Let $A\in M_{n}(R)$ and $f(x)$ be the characterestic polynomial of $A$. Is it true that $f'(x)=\sum_{i=1}^{^{n}}\sum_{j=1}^{n}\det(xI-A(i\mid j))$ which $A(i\mid j)$ is a submatrix of $A$ obtained by ...
3
votes
3answers
125 views

Minimal polynomial, determinants and invertibility

I need to prove: if a matrix $A$ is invertible, then the minimal polynomial $m_a(0) \neq 0$ There is one definition I am unsure of or need help making more clear. I will proceed with proof by ...
1
vote
1answer
114 views

Rank of a Vandermonde Matrix with additional weighted columns

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
3
votes
0answers
383 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 caculating them? So if I calculated determinant, minimal ...