6
votes
4answers
384 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 ...
0
votes
3answers
88 views

for which a, the matrix A is diagonalizable?

A = $ \begin{pmatrix} 2a+3 & 0 & 0 \\ -a-3 & a & a+3 \\ a & a & a+3 \\ \end{pmatrix} $ Characteristic polynomial: $ ...
3
votes
0answers
28 views

Determinant of a generalization of Moore matrices

The Moore matrix over $\mathbb{F}_q$ is the $n\times n$ matrix whose i'th row is: $a_i,a_i^q,a_i^{q^2},\dots,a_i^{q^{n-1}}$. The determinant of this matrix is the product of all linear combinations ...
0
votes
1answer
58 views

Equivalent condition for interpolation polynomial

Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ ...
1
vote
0answers
47 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
3answers
35 views

Evaluate the determinant

Let the following determinant, where $f_i$ is a polynomial with order of at most $n-2$. Evaluate the determinant: $$\left| {\begin{array}{*{20}{c}} {{f_1}({a_1})} & {{f_1}({a_2})} & {...} ...
4
votes
4answers
207 views

Question about Axler's proof that every linear operator has an eigenvalue

I am puzzled by Sheldon Axler's proof that every linear operator on a finite dimensional complex vector space has an eigenvalue (theorem 5.10 in "Linear Algebra Done Right"). In particular, it's his ...
2
votes
1answer
169 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 ...
4
votes
1answer
57 views

Determinant of a circulant matrix as Chebyshev-like recurrence

It is while studying the Hückel Method of Physical Chemistry that I came across the following recurrence relation: \begin{align*} U_n(x)=xU_{n-1}(x)-U_{n-2}(x)+(-1)^{n-1}(4+2x) \end{align*} Where ...
6
votes
1answer
140 views

Determinant vanishing over polynomial ring

Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
1
vote
1answer
87 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
1answer
69 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.
14
votes
2answers
171 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
97 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
72 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
45 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
73 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
427 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
460 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
136 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
81 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
328 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 ...
4
votes
1answer
294 views

Rank of a rectangular Vandermonde Matrix to which weighted columns are added

A Vandermonde matrix: $\left(\begin{array}{ccc} 1 & \alpha_{0} & \dots & \alpha_{0}^{n} \\ 1 & \alpha_{1} & \dots & \alpha_{1}^{n} \\ \vdots & \vdots & \ddots & ...
4
votes
1answer
947 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 calculating them? So if I calculated determinant, minimal ...
11
votes
3answers
347 views

Determinant of Abstract Matrix

Given an $n \times n$ matrix $A$, where $x$ is any real number: $A = \left[ \begin{array}{ c c c c c c c c } 1 & 1 & 1 & 1 & 1 & 1 & \cdots & 1 \\ 1 & x ...