# 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 $~T~$.

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• 145
49 views

### Intertwiner space isomorphism with invariante space

Let $V,W$ be two representation vector spaces of some group $G$. Thus, an intertwiner is as a linear map $T:V\rightarrow W$ satisfying $$T(g\cdot v)=g\cdot T(v)$$ We'll denote the space of ...
• 931
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### Linear algebra question about span [duplicate]

Prove or disprove the following. If S1 and S2 are arbitrary subsets of a vector space V , then the intersection of their spans (⟨S1⟩ ∩ ⟨S2⟩) equals the span of their intersection (⟨S1 ∩ S2⟩) .
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### Find subspace $T$ of $\mathbb{R}^{3}$ such that $\mathbb{R}^{3} =V \oplus T$

\begin{array}{l} V=\{( a+2b,2a+8b+2c,a+10b+4c) \ |\ a,b,c\in \mathbb{R}\}\\ =\{a( 1,2,1) +b( 2,8,10) +c( 0,2,4) \ |\ a,b,c\in \mathbb{R}\}\\ =Sp\{( 1,2,1) ,( 2,8,10) ,( 0,2,4)\} \end{array} Then I ...
• 13
1 vote
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### Existence of invariant subspaces (2)

Let $A$ be an $n\times n$ matrix with complex entries.If $n\ge 4$, which one of the following statements is true? $A$ does not have any invariant subspace in $\Bbb C^n$. $A$ has an invariant subspace ...
1 vote
40 views

### Question about how this implies an invariant subspace

The context of this is the proof of (part) of Schur's Lemma in Howard Georgi's Lie Algebras in particle physics. We want to prove theorem 1.3 which states "if $D_1(g)A = AD_2(g) \forall g \in G$ ...
• 143
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### Find a bijection between invariant subspaces of dimension m and invariant subspaces of dimension n-m

Let $V$ be a finite dimensional $K$-vector space of dimension $n$, and $T$ a linear transformation. I want to find a bijection between $T$-invariant subspaces of dimension $m$ and $T$-invariant ...
68 views

### A doubt in Fulton and Harris regarding invariant subspace

This is a doubt in the section 3.3, Induced representations of F&H. Let $G$ be a group and $H$ is a subgroup and $V$ is a representation of $G$ and $W \subset V$ is a subspace of $V$ which is $H$ ...
• 751
1 vote
76 views

### Group representation over $\mathbb{Z}$

The representation theory of finite groups over the fields $\mathbb{C}$ or $\mathbb{Q}$ is clear. I am wondering what happens if we consider something similar over the ring $\mathbb{Z}$. To be more ...
• 427
1 vote
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### General method for finding invariant subsapces of a nonlinear system

Suppose we are given a system: $$\dot{x_{1}} = f_{1}(x_{1},...,x_{n})$$ $$...$$ $$\dot{x_{n}} = f_{n}(x_{1},...,x_{n})$$ And are interested in finding subspaces of the vector space that are invariant ...
• 402
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### Integer vector subsets invariant under rational matrix

Let $A$ be an $n\times n$ matrix with rational coefficients. Define $H$ to be the maximal subset of $\mathbb{Z}^n$ such that $AH\subset H$. Question: How to describe $H$ in terms of $A$ (its Jordan ...
• 427
1 vote
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### Invariant Subspaces of $C_c^\infty(\mathbb{R})$

Does there exist a nontrivial subspace of $C_c^\infty(\mathbb{R})$ that is invariant under (horizontal) translation (i.e. any element of this subspace must also have its translates in the subspace)? I ...
68 views

### Isomorphism in quotient spaces of linear spaces

Let $E$ be a linear space and $V$ a linear subspace. No assumption is made on the dimension of $E$. We write $G = \{g \in GL(E) \mid g(V) = V\}$ where $GL(E)$ is the group of invertible linear maps of ...
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### Is every subspace can be represented as T-cyclic for some T? [closed]

Definition of $T$-cyclic subspace: Let $T$ be an linear operator on $V$. Take $v \in V$. The subspace generated by $\text{span}(\{v,T(v),T^2(v),\cdots\})$ is called $T$-cyclic subspace generated by $v$...
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• 2,183
1 vote
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• 323
1 vote
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### Commuting matrices and their Jordan forms

I am recently studying commuting matrices. I was reading the book Invariant Subspaces of Matrices with Applications(Godberg, Lancaster and Rodman) and pg.295-296 claims that a matrix $X$ is a solution ...
32 views

### Proving that every invariant subspace of $\mathbb{F}^n$ is of the form $span(e_1,...,e_k)$

Let $T_{J_n(0)}:\mathbb{F}^n \to \mathbb{F}^n$. I need to prove that every invariant subspace $W$ of $\mathbb{F}^n$ is of the form $Span(e_1,...,e_k)$ for some $0\leq k \leq n$. This is what I've got ...
• 323
228 views

### Connection between invariant subspaces and eigenvectors of a linear operator: Showing that if $A,B$ commute then $A$ and $B$ share an eigenvector [duplicate]

I got stuck trying to show that if $A,B$ are two linear operators on a finite dimensional vector space $V$ over the field $\mathbb{F}$, then $A$ and $B$ share an eigenvector. Quick Googling revealed ...
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• 31
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### If $T$ has irreducible minimal polynomial, can one decompose the space as a direct sum of invariant subspaces without non-trivial invariant subspaces.

$T: W \rightarrow W$ is a linear operator on a finite dimensional vector space $W$. If the minimal polynomial of linear operator $T$ on $W$ is irreducible, is there a way to decompose $W$ into W=W_1\...
22 views

• 37
1 vote
44 views

### Finding invariant factors of a matrix given a polynomial which annihilates the matrix

We have a square matrix A with entries from $Q$ and a polynomial $g(x) = (x^2 +3x+5)(x−1)^2(x^3 +1)$ $g \in Q[x]$ Such that g(A) is 0. The minimal polynomial is said to be of the degree 2 and we are ...
67 views

### Two dimensional invariant subspace

Let $T:\mathbb{R}^3\rightarrow \mathbb{R}^3$ be a linear transformation defined by $T(x,y,z)=(x+y,y+z,z+x)$. I want to find two dimensional invariant subspace under $T$. I know that two dimensional ...
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### Let $f\in L^2(\Bbb{T}),\ f\ne0$ and $M=\overline{f\wp_+}$. Prove that, $M$ is simply invariant for $\chi_1$ iff $f\notin \chi_N M$ for some $N>0$
Here we denote $\chi_n$ to be the functions on $\Bbb{T}$ defined as $\chi_n(z)=z^n\ \forall z\in\Bbb{T}$ for $n\in\Bbb{Z}$. We define $\wp_+=\{\sum\limits_{n=0}^N a_n\chi_n|\ a_n\in\Bbb{C},N\ge0\}$ be ...