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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 $~T~$.

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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\...
fool's user avatar
  • 297
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0 answers
18 views

A direct sum decomposition [duplicate]

I have tried to solve this problem. I am studying for an exam. Let $V$ be a real finite-dimensional vector space and let $T \colon V \rightarrow V$ be and operator such that for some positive integer $...
user123456's user avatar
3 votes
1 answer
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 ...
Powder's user avatar
  • 931
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0 answers
5 views

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⟩) .
muraleetharan kugaram's user avatar
0 votes
2 answers
59 views

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 ...
Shai's user avatar
  • 13
1 vote
0 answers
70 views

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 ...
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1 vote
1 answer
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 $D_1(g)A = AD_2(g) \forall g \in G$ ...
JohnA.'s user avatar
  • 143
0 votes
0 answers
28 views

$A$-invariant subspace of $\mathbb{F}^n$

Let $\mathbb{F}$ be a field and $A\in M_n(\mathbb{F})$. Let the minimal polynomial of $A$ be $\prod_{i=1}^m q_i(x)^{\ell_i}$, where each $q_i(x) \in \mathbb{F}[x]$ is monic irreducible, each $\ell_i&...
MinaaaaaniM's user avatar
-1 votes
1 answer
64 views

A problem about diagonalize invariant subspaces [closed]

Let $V$ be a non-zero finite-dimensional vector space, A belongs to End($V$). Also, for any invariant subspace $M$ of A, there exists an invariant subspace $N$ of A such that $V=M\oplus N$. Prove: A ...
淘宝者's user avatar
2 votes
1 answer
37 views

Show that $Z(v,f) \oplus Z(w,f) = Z(v+w, f)$ if minimal polynomials are coprime.

I am currently learning Linear Algebra and just can't finish this one exercise, at least partially... We are given some field $F$ and the normal vector space $V = K^n$. By $Z(A,v)$ we denote $\left\...
user avatar
2 votes
1 answer
76 views

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 ...
Juan Claver's user avatar
0 votes
1 answer
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 $H$ ...
Eloon_Mask_P's user avatar
1 vote
0 answers
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 ...
QMath's user avatar
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1 vote
1 answer
73 views

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 ...
Mani's user avatar
  • 402
0 votes
0 answers
60 views

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 ...
QMath's user avatar
  • 427
1 vote
2 answers
119 views

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 ...
Morgan Zariski's user avatar
3 votes
1 answer
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 ...
mathcounterexamples.net's user avatar
2 votes
1 answer
91 views

Showing that $\phi$ is irreducible iff $\psi(g,h): X \to \phi(g)X\phi(h^{-1})$ is

We have a matrix representation $\phi$ of degree $d$ of a finite group $G$ over a field $F$, where character of $F$ doesn't divide $G$. From it we construct a representation $\psi : G \times G \to Aut(...
TdotA's user avatar
  • 400
3 votes
0 answers
98 views

Can you verify my proof for this result about eigenvalues of the quotient operator?

This is problem 35 of section 5.A of Axler's book Linear Algebra Done Right (3rd Edition). The problem goes as follows: Suppose $V$ is finite-dimensional, $T$ is a operator on $V$, and $U$ is ...
armoredchihuahua's user avatar
0 votes
1 answer
82 views

Matrix with invariant subspace

Find the 5×5 real matrix $ A$ such that: $ A(1\ 1\ 1\ 1\ 1)^T =(1\ 1\ 1\ 1\ 1)^T $ $ A^{-1} = A^T $ $ A^6 = A $ $ A $ has distinct eigenvalues. If $ W $ is the subspace generated by $ (1\ -1\ 0\ 0\ 0)...
newyjsk's user avatar
2 votes
0 answers
49 views

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 $v$...
Ganesh Karthi's user avatar
0 votes
0 answers
322 views

Did I solve correctly?

Q: Let $f:V\rightarrow V$ be a linear operator on real vector space $V$. Prove if dim$V\geq2$, then there exist invariant subspace of $V$, with dimension $2$. My approach: Induction: Basis - if dim$V=...
Chess player's user avatar
1 vote
0 answers
38 views

Finding an invariant subspace for a relative of the Volterra operator

I want to find a (non-trivial, closed) invariant subspace for the operator $T: C[0,1] \rightarrow C[0,1]$ defined by $(Tf)(x) = \displaystyle\int_0^x2tf(t)dt + f(1)$ The usual candidates don't seem to ...
Gollol's user avatar
  • 58
0 votes
0 answers
37 views

Maximal elements of subspaces

Let $W$ be a subspace of a vector space $V$ and $f$ be an endomorphism of $V$. Show that $$X = \{U \subseteq W\,|\,U \text{ is an $f$-invariant subspace of V}\}$$ with the partial order $\subseteq$ ...
Sheep's user avatar
  • 53
2 votes
1 answer
66 views

Does the subspace $W'$ exist such that $\mathbb{R}^3=W\oplus W'$?

Let $F=\begin{pmatrix} 2 & -1 & 0\\ 1 & 0 & 0\\ -1 & 1 & 1\end{pmatrix}$, show that $W$ is an invariant subspace under $F$, with $W=span\{(1,1,1),(1,1,2)\}$. Does an invariant ...
Acedium 20's user avatar
1 vote
1 answer
53 views

Finding an $\mathcal{A}$-invariant subspace of $\mathbb{R}^2$

There is a linear operator $\mathcal{A}:\mathbb{R}^2\mapsto \mathbb{R}^2$, such that $\mathcal{A}(x,y)=(x,x+y)$. Find all $\mathcal{A}$-invariant subspaces of $\mathbb{R}^2$. Let $S$ be some ...
bb_823's user avatar
  • 2,183
4 votes
2 answers
109 views

Is Sequence Convergence a topological Invariant?

I am stuck on a question that goes like this: Let $(X, \mathcal{T})$ be a topological space, and let $A \subseteq X$. $A$ is called sequentially closed if the limit point of every convergent sequence ...
Ryukendo Dey's user avatar
0 votes
1 answer
48 views

Confused regarding Linear Invariant Subspaces

Assuming a linear transformation $T : \mathrm{V \rightarrow V}$, let $\mathrm{W \subset V}$ so that $\mathrm{W} \ne \{0\}$. If $T(\mathrm{W}) \subseteq \mathrm{W}$ then $\mathrm{W}$ is $T$ invariant, ...
Baconface's user avatar
0 votes
1 answer
40 views

Prove that $W$ is invariant subspace of $\mathbb{R}^4$ with respect to linear operator $\mathcal{T}$

During the lecture I've missed out on, there was the following problem: Let $\mathcal{T}:\mathbb{R}^4\to\mathbb{R}^4$, where linear operator $\mathcal{T}$ is defined by $\mathcal{T}(a,b,c,d)=(a+b+2c-...
bb_823's user avatar
  • 2,183
1 vote
0 answers
42 views

Direct sum of primary decompositions vector spaces intersected with a T invariant subspace.

Here is the question: I am stuck on this question for more than a few hours. All my attempts to solve this question failed. What I tried to do: Show the following equality: $W = W \cap V = W \cap (...
Emanuel L's user avatar
2 votes
0 answers
54 views

Finding all invariant subspaces of $T_A:\mathbb{C}^n \to \mathbb{C}^n$ where $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)$

Let $A=J_{n_1}(\lambda_1)\oplus ... \oplus J_{n_k}(\lambda_k)\in \text{Mat}_n(\mathbb{C})$ when $\lambda_i\neq\lambda_j$ for every $i\neq j$. I need to describe all the $T$-invariant subspaces of $T=...
Staltus's user avatar
  • 323
1 vote
1 answer
149 views

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 ...
Ahmet Sakal's user avatar
0 votes
1 answer
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 ...
Staltus's user avatar
  • 323
0 votes
2 answers
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 ...
Cartesian Bear's user avatar
0 votes
1 answer
36 views

Equivalent conditions for reducibility of $\mathbb{C}$-linear operator on ${\mathbb{C}}^n$

Let $A$ be a $\mathbb{C}$-linear operator from ${\mathbb{C}}^n$ to itself, with rank$A=p < n$. For complex linear subspace $E$ of ${\mathbb{C}}^n$, $A$ is called reduced by $E$ if both $E$ and $E^{\...
Kiyoon Eum's user avatar
0 votes
0 answers
56 views

Let $U=\langle(1,0,1)\rangle$. Determine the equations of the orthogonal subspace of $U$

Let $U = \langle(1,0,1)\rangle$. Determine the equations of the orthogonal subspace of $U$ and an orthonormal basis of it. Obtain an expression for the vector $v = (-8, 3, 5)$ of the form $v = x+ y$, $...
Lucía's user avatar
  • 31
0 votes
1 answer
164 views

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\...
Gunt Ryumet's user avatar
0 votes
0 answers
22 views

Show that there exists exactly one $l_A$-invariant $K$-subspace of dimension $1$ in $K^2$.

So I have this exercise where I want to check my solutions. Can someone help me? Let $A = \begin{pmatrix} a &b \\ 0& d \end{pmatrix} ∈ Mat_{2,2}(K)$ with $a, b, d ∈ K$, where $a ≠ 0$ and $d ≠ ...
Marco Di Giacomo's user avatar
3 votes
1 answer
70 views

Spanning set of symmetric invariants of tensor powers

Let $M$ be a module over a commutative ring $R$ and let the symmetric group $\Sigma_n$ act on $M^n$ and $M^{\otimes n}$ by permuting factors. For $v \in M^n$, let $\text{Stab}(v)$ denote the ...
deaton.dg's user avatar
  • 212
1 vote
0 answers
58 views

Exercise 1, Section 6.7 of Hoffman’s Linear Algebra

Let $E$ be a projection of $V$ and let $T$ be a linear operator on $V$. Prove that the range of $E$ is invariant under $T$ if and only if $ETE=TE$. Prove that both the range and null space of $E$ are ...
user264745's user avatar
  • 4,227
0 votes
0 answers
85 views

direct sum of linear bounded operators

I have a question on (orthogonal) direct sums of an Operator. In particular, I was wondering if the direct summands of a linear and bounded operator $T$ on a complex Hilbertspace $H$ are all $T$-...
BabyWienerSpace's user avatar
2 votes
0 answers
44 views

pure m-Isometries

I am currently studying a paper by Jim Agler and Mark Stankus called m-Isometric Transformations of Hilbertspaces which you can find here https://core.ac.uk/download/pdf/19158531.pdf. My question is ...
BabyWienerSpace's user avatar
1 vote
1 answer
38 views

Reducible Subspaces

I have a question regarding reducing subspaces: Let $H$ be a complex Hilbertspace and let $T$ be a bounded linear operator on $H$. Furthermore, let $H_1$ be a $T$-reducing closed subspace, which means ...
BabyWienerSpace's user avatar
1 vote
1 answer
128 views

Exercise 2, Section 6.4 of Hoffman’s Linear Algebra

Let $W$ be an invariant subspace for $T$. Prove that the minimal polynomial for the restriction operator $T_W$ divides the minimal polynomial for $T$, without referring to matrices. My attempt: We ...
user264745's user avatar
  • 4,227
0 votes
0 answers
45 views

Showing that a subspace is invariant under a representation r

I'm having trouble with a problem from Yvette Kosmann-Schwarzbach's book called Groups and Symmetries. Let $V=\{(z_1,z_2,z_3)\in \mathbb{C^3}|z_1+z_2+z_3=0 \}$ a vector subspace of $\mathbb{C^3}$ and $...
pieq3's user avatar
  • 37
1 vote
0 answers
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 ...
Tushita Pandey's user avatar
0 votes
0 answers
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 ...
LoveMath's user avatar
  • 117
2 votes
1 answer
86 views

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 ...
MathBS's user avatar
  • 3,144
0 votes
0 answers
28 views

Can $\lambda_i$'s root subspace be divided into several eigenspaces direct sum?

Let $W_{\lambda_i}$ be $\lambda_i$'s root subspace of linear transformation $\mathscr{A}$. I just learned $W_{\lambda_i}$ can be represented as direct sum of some cyclic subspaces such as $W_{\...
P. Scotty's user avatar
  • 115
1 vote
0 answers
75 views

About running variance in Batch Normalization layer

A batch normalization(BN) layer is normally used to reduce the covariance shit problem in neural networks. Where in a layer, input $x$ will be normalized to something like $x^{\prime}$ = $\frac{x-\mu(...
Dennis Wu's user avatar

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