Questions tagged [characteristic-polynomial]
The characteristic polynomial of a square matrix is a polynomial that is invariant under matrix similarity and has the eigenvalues as roots.
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Is there a simple way to derive left eigenvectors from right eigenvectors in the case of a non-linear eigenvalue problem?
First I’ll recap the normal eigenvalue problem to help explain what I’m asking. Say we have an $n\times n$ matrix $A$. Then $\det(\lambda I-A)$ is its characteristic polynomial and its zeroes are the ...
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Proving that a matrix and its transpose have the same characteristic polynomial [duplicate]
I'm working on a linear algebra problem about characteristic polynomials and would appreciate some guidance.
For any square matrix $A$, prove that $A$ and $A^T$ have the same
characteristic ...
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Solving the recurrence $ a(n)=a(n-4)+\left \lfloor{\frac{(n+3)^2+8}{24}}\right \rfloor $, with $a(1)=a(2)=a(3)=1$ and $a(4)=2$
I need help solving the recurrence relation
$$
a(n)=a(n-4)+\left \lfloor{\frac{(n+3)^2+8}{24}}\right \rfloor
$$
with starting values $a(1)=a(2)=a(3)=1$ and $a(4)=2$.
I am looking for closed form ...
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The characteristic polynomial of Householder matrix
Some time ago, I saw an interesting way to obtain characteristic polynomial of Householder matrix here. Unfortunately, I can't understand how to get this formula by characteristic polynomial ...
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Prove $\frac{1}{det(1 - \alpha t)} = \exp(\sum_{r=1}^\infty tr(\alpha^r) \frac{t^r}{r})$
This is exercise VI.6.12 from Aluffi's Algebra: Chapter 0.
The full problem statement and hint is:
Let $\alpha$ be a linear transformation of a finite-dimensional $\Bbb{C}$-vector space $V$. Prove ...
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A largest eigenvalue $\lambda_0$ and the others from a specific characteristic equation
My book says that
the following characteristic equation
$$\sum_{i=1}^n\frac{p_i \lambda}{1+p_i \lambda} = 1 \space\space, \space\space where \space 0 < p_i < 1 $$
"has a unique positive ...
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Recurrence for characteristic polynomial coefficients
I am trying to find the characteristic polynomial of a $6\times 6$ block matrix $A$, where each block is a $\ell\times\ell$ matrix, resulting in a $6\ell\times 6\ell$ matrix. The matrix $A$ is defined ...
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Product of two non similar matrices
Consider two matrices $A_{3\times2}$ and $B_{2\times3}$ then we can show that $\det(AB)=0$, and in fact $\det(P_{m\times n}Q_{n\times m})=0$ for all $m>n$, though what I am interested in is in the ...
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Characteristic polynomial of $\phi$ is a power of an irreducible polynomial
I came across this exercise and honestly I have no idea on where to start:
Let $V$ be a finite dimensional vector space over a field $k$. Let End$_k(V)$ be the ring of all linear transformations from $...
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Jordan form of a companion matrix
Let $n \ge 2$ be an integer. For $i = 0, 1, \dots , n-1$, let $c_{i} = {n \choose i}$. Find the Jordan form of the following companion matrix.
$$ A = \begin{pmatrix}
0 & 0 & 0 & \cdots &...
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Is there a smart way to compute the characteristic polynomial of this matrix?
I want to compute in a smart way the characteristic polynomial of this $3 \times 3$ matrix
$$ A_n = \begin{pmatrix}
n-4 & n-3 & n-2 \\
n-1 & n & n+1 \\
...
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Characteristic polynomial of inverse of A [duplicate]
How to express the characteristic polynomial of the Inverse of A in Terms of A? can i just use 1/det(A) = det(A^-1)
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General Formula for the coefficients of a characteristic polynomial
Let $A\in K^{n\times n}$, with $n \in \mathbb{N}$ then by
$$
\chi_A=\det(x\text{I}_n-A)=\sum\limits_{k=0}^{n}a_kx^k\in K[x]
$$
we define the characteristic polynomial of $A$.
I already know the ...
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Prove statements regarding commutator matrix $ AB - BA $
Let $n \in \mathbb{N}^{*}$ and $ A, B \in \mathcal{M}_{2n+1}(\mathbb{C}) $ such that $ A^{2}B^{2} = (AB)^{2} $. Prove that:
a) $ \det(AB-BA) = 0 $
b) If $ n=1 $ then $ (AB-BA)^{3} = O_{3} $
For a) I ...
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Motivation behind characteristic equations in recurrence relations
In learning about solutions to linear recurrences of the form $a_n = ia_{n-1}+ ja_{n-2}$, where $i, j$ are known constants, I came across the technique of using characteristic equations. I was ...
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