# Questions tagged [generalized-eigenvector]

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48 questions
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### Eigenvectors and eigenvalues relational proofs

Let A ∈ R^ n×n How do I prove that "If A has a finite number of distinct eigenvectors then each eigenvector must have a distinct eigenvalue." If A a symmetric matrix in R n×n . A is called positive ...
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### Trace in a generalized eigenspace

I am reading a paper and there is a steep in a proof that I cannot understand. The proof says: Suppose $v_\lambda$ is a generalized eigenvector corresponding to an eigenvalue $\lambda$ of the map $T$...
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### Finding the Jordan Form of a matrix…

I know that this type of question has been asked on here before but I am still having a hard understanding what is going on. The text that I am learning from is "Linear Algebra Done Right by Sheldon ...
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### Decomposition of invariant subspaces

Consider a set of $d$ linearly independent generalized eigenvectors of some matrix $A \in \mathbb{C}^{d \times d}$. Suppose this set is decomposed into $M$ distinct Jordan chains and the $m$-th Jordan ...
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### What is the simplified form of the generalized eigen space when the characteristic polynomial does not split in the given field.

Let $V$ be a finite dimensional vector space over a field $\mathbb{F}.$ Let $T$ be a inear operator on $V$ and $\lambda \in \mathbb{F}$ be an eigenvalue of $T$ of algebraic multiplicity $m.$ Now ...
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### Are they similar matrix

Do $\begin{bmatrix} 0&i&0\\0&0&1\\0&0&0 \end{bmatrix}$ and $\begin{bmatrix} 0&0&0\\-i&0&0\\0&1&0 \end{bmatrix}$ are similar.Is this True/false ...
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### Show that a is the length of the biggest Jordan chains corresponding to an eigenvalue.

Let V be a fi nite-dimensional vector space, T belong to L(V) and T has an eigenvalue x with the corresponding space of generalized eigenvectors Ux. Let a = min{k : (T-x)^k|Ux = 0}. Show that a is the ...
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### Let $f$ and $g$ be two linear operators, show that eigenvalues of $g$ are of the form $a, a-1, a-2, \ldots, a - (n - 1)$?

Let $f$ and $g$ be two linear operators with $\dim(L) = n$, where $L$ is a vector space over a field $F$ with characteristic zero. Assume that $f^{n} = 0$, $\dim \ker(f) = 1$, and $gf-fg = f$. How it ...
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### Let $T:V\to V$ be a diagonalizable linear map on a vector space $V$. Does it hold that $T^{**}:V^{**}\to V^{**}$ is diagonalizable?

Edit: I found my own answer. But I will be happy to see other answers as well. Question: Let $T:V\to V$ be a linear operator on a vector space $V$. (a) Suppose that $T$ is diagonalizable. Does ...
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### Generalized Eigenspaces Associated to Different Values

Let $V$ be some vector space, and $T \in \mathcal{L}(V)$. If $a \neq b$, then $G(a,T) \cap G(b,T) = \{0\}$, where $G(b,T)$ denotes the generalized eigenspace. I am having a lot of trouble with this ...
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### Dual formulations of Generalized Eigenvectors

I've seen two definitions of generalized eigenvectors. One is $(A−λ⋅I)^kv=0$ and one where it's $Av = \lambda B v$. I get that they are the same thing but I'm curious what is the point of both ...
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### When will there exist a basis of eigenvectors?

From what I have found, "there exists a basis of eigenvectors if and only if the algebraic and multiplicity of each eigenvalue is the same." However, I cannot relate the multiplicity of eigenvalues to ...
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### Is $\text{dim}\, G(T_\mathbb C ,\lambda)=\text{dim}\, G(T,\lambda)$ when $\lambda$ is real?

Is $\text{dim}\, G(T_\mathbb C ,\lambda)=\text{dim}\, G(T,\lambda)$ when $\lambda$ is real? I am studying the theorem 9.23(c) of the S.Axler's Linear Algebra Done Right (3rd Edition) on page 284 of ...
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### Generalized eigenvectors for Jordan canonical form (and theory)

I am learnnig how to find a Jordan normal form of a matrix, and I got stuck on four items. Can you please explain to me and check if I am right with my understanding? (I do not have a real example on ...
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### Can a polynomial with degree (r+s-1) have (r+s) distinct roots?

While reading about generalised eigenvectors, i came to a strange proof. It goes like If p is a polynomial which has r distinct roots {λ1,λ2...,λr}, and q is a polynomial with s distinct roots {μ1,μ2....
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### How to find generalized Eigen vectors of a matrix with Eigen vectors already on diagonal?

I have a matrix A= \begin{bmatrix} 1 & -1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & -1\\ 0 & 0 & 1 & 1 \end{bmatrix} This matrix has ...