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Questions tagged [invariant-subspace]

This tag is for questions relating to "Invariant Subspaces". Mathematically, an invariant subspace of a linear mapping $~T : V → V~$ from some vector space $~ V~$ to itself is a subspace $~W~$ of $~V~$ that is preserved by $~T~$.

Definition: Suppose that $~T:V\to V~$ is a linear transformation and $~W~$ is a subspace of $~V~$. Suppose further that $~T(w)\in W~$ for every $~w\in W~$. Then $~W~$ is an invariant subspace of $~V~$ relative to $~T~$.

  • The subspaces $\operatorname{null}(T)$ and $\operatorname{range}(T)$ are invariant subspaces under $T$.
  • $\{0\}$ and $V$ are trivial invariant subspaces.
  • We do not have any special notation for an invariant subspace, so it is important to recognize that an invariant subspace is always relative to both a superspace $(V)$ and a linear transformation $(T)$, which will sometimes not be mentioned, yet will be clear from the context.
  • The linear transformation involved must have an equal domain and codomain — the definition would not make much sense if our outputs were not of the same type as our inputs.

References:

https://en.wikipedia.org/wiki/Invariant_subspace

https://math.okstate.edu/people/binegar/4063-5023/4063-5023-l18.pdf

http://alpha.math.uga.edu/~pete/invariant_subspaces.pdf

419 questions
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If a linear map $T$ has a $k$-dimensional invariant subspace, does it admit an $n-k$ invariant subspace?

Let $V$ be an $n$-dimensional real vector space, and let $1
noname1014
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Can a nonconstant function that is invariant to this affine transformation exist?

I'm trying to come up with an example of a nonconstant function $f:\mathbb{R}^2\to\mathbb{R}$ such that $f(x, y)=f(A(x), A(y))$ for any affine transformation of the form $A(z) = az+b$ where $a>0, b\in\mathbb{R}$. Intuitively, I don't think it…
noob
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For $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times k}$, does $(I_{n}-aA)^{-1}Bb$ determine $a\in\mathbb{R}$ and $b\in\mathbb{R}^k$?

Let $A$ be a real $n\times n$ matrix, and $B$ be a real $n\times k$ matrix of rank $k
mark
  • 594
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Invariant subspaces of Lie group vs invariant subspaces of Lie algebra

I am starting to study infinite-dimensional representations of Lie groups and I am wondering about the following: Let $G$ be a connected Lie group with Lie algebra $\mathfrak g$ and with a representation $\Phi: G \to \text{GL}(V)$, with $V$…
Mekanik
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Existence of $n-1$ dimensional invariant subspace of $V$ over $\mathbb{R}$ given characteristic polynomial has a real root.

$V$ is a finite dimensional vector space over $\mathbb{R}$ with $\dim V \ge 1$ and $\phi \in L(V, V)$ is an endomorphism. Its characteristic polynomial $w_{\phi}(\lambda)$ has a real root. Prove the existence of an $n-1$ dimensional invariant…
user4201961
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Does the action of a linear map on $k$-dimensional subspaces determine it up to scaling?

Let $V$ be a real $d$-dimensional vector space, and let $1 \le k \le d-1$ be a fixed integer. Let $A,B \in \text{Hom}(V,V)$, and suppose that $AW=BW$ for every $k$-dimensional subspace $W \le V$. Is it true that $A=\lambda B$ for some $\lambda \in…
Asaf Shachar
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Every linear operator on $\mathbb{R}^5$ has an invariant 3-dimensional subspace

I am trying to determine whether the following statement is true or false: Every linear transformation on $\mathbb{R}^5$ has an invariant 3-dimensional subspace. Since $\dim(\mathbb{R}^5)=5$ then given any linear operator $T$ on $\mathbb{R}^5$ I…
Sarah
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Is there a connection between my density formula and an invariant mean defined by a folner sequence of rational numbers?

I am a first-year undergraduate who stumbled upon natural density. I am working on extending this definition to the subset of rational numbers. While most people would wait until they are older, I am already attempting to solve the problem. I sent a…
Arbuja
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Invariant subspace problem for $\ell^2(\mathbb{N})$

Is the invariant subspace problem known for $\ell^2(\mathbb{N})$ or for more general $L^2$ spaces, i.e. does every bounded linear operator $T \colon \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N})$ have a non-trivial (closed) T-invariant subspace?
Zahlenteufel
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Invariant subspace of $T$ (normal) is also an invariant subspace of $T^\ast$.

I am struggling with the following question Let $V$ be a finite dimensional vector space and $T:V\rightarrow V$ be a linear normal operator ($T^\ast T = TT^\ast$), and $W$ an invariant subspace of $T$ ($T(W)\subseteq W)$. Prove $W$ is also an…
Theorem
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$T \in \mathcal L (V)$ has no real eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension.

Suppose $V$ is a real vector space and $T \in \mathcal L (V)$ has no real eigenvalues. Prove that every subspace of $V$ invariant under $T$ has even dimension. Solution : Suppose $U$ is a subspace of $V$ that is invariant under $T$. If $\dim U$…
Maggie
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Hermitian Operators and the Spectral Theorem

I understand that in a finite-dimensional vector space $V$, a diagonalizable linear operator $T: V \to V$ decomposes $V$ into a direct sum of its invariant eigenspaces, on each of which it restricts to a scalar multiple of a projection. The Spectral…
David Grabovsky
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Non-orthogonal invariant subspaces

Let $\Gamma\subset\mathrm O(\Bbb R^n)$ be a finite group of orthogonal matrices. Let $U_1,U_2\subseteq\Bbb R^n$ be two irreducible invariant subspaces w.r.t. $\Gamma$ with $U_1\cap U_2=\{0\}$, which are not orthogonal to each other, i.e. there are…
M. Winter
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Any non-trivial $T$-invariant subspace of $V$ contains an eigenvector of $T.$

Let $T$ be a linear operator on a finite-dimensional vector space $V$. Deduce that if the characteristic polynomial of $T$ splits, then any non-trivial $T$-invariant subspace of $V$ contains an eigenvector of $T.$ Let $W$ be a $T$-Invariant…
user464147
5
votes
1 answer

Finding all invariant subspaces of a matrix

Let $\alpha$ be a real number. Find all invariant subspaces for the matrix $$ \begin{pmatrix} \cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}. $$ How does the result depend on $\alpha$? I am…
Johnny44
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