# Tagged Questions

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### 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 ...
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### 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 ...
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### Determinant of a circulant matrix as Chebyshev-like recurrence

It is while studying the HÃ¼ckel 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 ...
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### 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 ...
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### 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 ...
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### 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 $a$ ...
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### 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 $\omega = (\omega_1,\cdots,\omega_n)$ ...
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### 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 $x$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...