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 $~T~$.
392
questions
-1
votes
0
answers
34
views
Diagonalization of linear transformation [duplicate]
Let $V$ be a finite dimensional vector space over $\mathbb{C}$ and let $T:V\rightarrow V$ be a linear transformation such that any subspace of $V$ which is stable under $T$ has a complement which is ...
2
votes
1
answer
60
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(...
- 381
3
votes
0
answers
67
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 ...
0
votes
1
answer
73
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)...
- 1
2
votes
0
answers
33
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 $v$...
0
votes
0
answers
141
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=...
- 112
1
vote
0
answers
27
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 ...
- 44
0
votes
0
answers
30
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 $\subseteq$ ...
- 53
2
votes
1
answer
60
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 ...
- 131
1
vote
1
answer
35
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 ...
- 1,703
4
votes
2
answers
88
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 ...
- 433
0
votes
1
answer
44
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, ...
0
votes
1
answer
31
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-...
- 1,703
1
vote
0
answers
39
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 (...
- 45
2
votes
0
answers
48
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=...
- 301
0
votes
1
answer
55
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 ...
- 26
0
votes
1
answer
30
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 ...
- 301
0
votes
0
answers
47
views
Irreducible representations of the tensor product of three Lie groups and projectors of an invariant Hermitian matrix.
Let $C^{2}$ be an irreducible representation of the Lie group $SU(2)$. Let us consider the tensor product of three copies of the representation. Then $C^{2}\otimes C^{2}\otimes C^{2}=2C^2\bigoplus C^4$...
0
votes
2
answers
61
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 ...
- 725
0
votes
1
answer
30
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^{\...
- 348
0
votes
0
answers
53
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 $v = x+ y$, $...
- 21
0
votes
1
answer
120
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 $$W=W_1\...
- 87
0
votes
0
answers
15
views
Show that there exists exactly one $l_A$-invariant $K$-subspace of dimension $1$ in $K^2$.
So I have this exercise where I want to check my solutions. Can someone help me?
Let $A = \begin{pmatrix} a &b \\ 0& d \end{pmatrix} ∈ Mat_{2,2}(K)$ with $a, b, d ∈ K$, where $a ≠ 0$ and $d ≠ ...
- 694
0
votes
0
answers
30
views
Boundary conditions and invariant sets for PDE's
I'm looking into invariance set theorems for parabolic PDE's (such as https://memoriam.cse.umn.edu/hfw/Math/InvariantSets.pdf and https://arxiv.org/pdf/0911.4526.pdf). However, they don't specify ...
- 43
3
votes
1
answer
51
views
Spanning set of symmetric invariants of tensor powers
Let $M$ be a module over a commutative ring $R$ and let the symmetric group $\Sigma_n$ act on $M^n$ and $M^{\otimes n}$ by permuting factors. For $v \in M^n$, let $\text{Stab}(v)$ denote the ...
- 212
0
votes
0
answers
45
views
Exercise 4, Section 6.7 of Hoffman’s Linear Algebra
Let $T$ be a linear operator on $V$. Suppose $V=W_1\oplus … \oplus W_k$,where each $W_i$ is invariant under $T$. Let $T_i$ be the induced (restriction) operator on $W_i$.
(a) Prove that $\det (T)=\det ...
- 3,619
1
vote
0
answers
46
views
Exercise 1, Section 6.7 of Hoffman’s Linear Algebra
Let $E$ be a projection of $V$ and let $T$ be a linear operator on $V$. Prove that the range of $E$ is invariant under $T$ if and only if $ETE=TE$. Prove that both the range and null space of $E$ are ...
- 3,619
0
votes
0
answers
63
views
Theorem 10, Section 6.7 of Hoffman’s Linear Algebra
Theorem 9: $V=W_1\oplus …\oplus W_k$$\iff$$\exists E_1,…,E_k\in L(V,V)$ such that
(i) each $E_i$ is projection ($E_i^2=E_i$)
(ii) $E_iE_j=0$, if $i\neq j$
(iii) $I=E_1+…+E_k$
(iv) $R_{E_i}=W_i$.
...
- 3,619
0
votes
0
answers
50
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$-...
- 330
2
votes
0
answers
30
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 ...
- 330
1
vote
1
answer
31
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 ...
- 330
1
vote
1
answer
110
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 ...
- 3,619
0
votes
0
answers
47
views
How to construct translation invariant distribution
There are serveral papers talk about how to construct a translation invariant distribution, however I couldn't get the idea of why those methods work and why we need it. If possible, could please give ...
- 11
0
votes
0
answers
39
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 $...
- 37
1
vote
0
answers
35
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 ...
0
votes
0
answers
9
views
Determine those s ∈ $\mathbb R$ for which $U_1$ + $T_s^+$ matches with $\mathbb R^\mathbb R$
For any s ∈ $\mathbb R$ the subsets of $\mathbb R^\mathbb R$ are difined:
$T_s^+$ := {f ∈ $\mathbb R^\mathbb R$ | f(x) = 0 for all x $\ge$ s},
$T_s^-$ := {f ∈ $\mathbb R^\mathbb R$ | f(x) = 0 for all ...
- 39
0
votes
0
answers
47
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 ...
- 97
2
votes
1
answer
76
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 ...
- 3,053
0
votes
0
answers
24
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_{\...
- 113
0
votes
0
answers
35
views
The eigenvalue of projection Matrix
Given a full rank symmetric matrix $H \in \mathbb{R}^{n \times n}$, and we generate two orthnormal vectors, say $v_1$, $v_2$. Let $V_2 = [v_1, v_2] \in \mathbb{R}^{n \times 2}$, then $V_2V_2^\top$ is ...
- 61
1
vote
0
answers
45
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(...
- 11
2
votes
1
answer
241
views
Exercise $6$, Section $5.A$ - Linear Algebra Done Right
Exercise: Prove or give a counterexample: if $V$ is finite-dimensional and $U$ is a
subspace of $V$ that is invariant under every operator on $V$, then $U = \{0\}$
or $U = V$.
Operator: The term ...
- 3,514
2
votes
2
answers
108
views
Admissible subspace is invariant?
Let $T$ be a linear operator on a vector space $V$. A subspace $W$ is called $T$-admissible if
$W$ is $T$-invariant
if $f(T)\beta$ lies in $W$, then there exists a vector $w$ in W, such that $f(T) \...
- 483
0
votes
1
answer
57
views
Block diagonalization with similiarity transform using invariant subspaces
How can I show that if the full n-vector space V can be written as a direct sum of subspaces V_i for i=1,... k, such that all V_i are invariant subspaces of diagonalizable matrix A, I can block ...
1
vote
1
answer
103
views
Let $V$ be a vector space over $\Bbb R$ of dimension $n$, and $T \colon V \to V$ be a linear trasformation. Choose the correct answer.
Let $V$ be a vector space over $\Bbb R$ of dimension $n$, and $T \colon V \to V$ be a linear trasformation. Choose the correct answer.
(a) There exist subspaces $V := V_0 \subset V_1 \subset V_2 \dots ...
- 920
2
votes
1
answer
62
views
Eigenvalues of invariant subspaces
Let $V$ be a vector space over $F$ and $f:V\mapsto V$ a linear map. If $dimV=n \geq 2$ and the only invariant subspaces of $V$ are $V$ itself and {$0_V$} ,then investigate if $f$ has eigenvalues.
I'm ...
- 345
0
votes
0
answers
52
views
2-dimensional invariant subspace using Jordan matrix
If given a matrix which is invertible, and knowing the Jordan canonical form is:
$$J = \begin{bmatrix} -2 & 0 & 0 & 0\\ 0&5&0&0\\0&0&-8&0\\0&0&0&8\end{...
- 33
2
votes
1
answer
137
views
existence of non-zero proper T-invariant subspace
Let $V$ be a finite dimensional vector space and $T:V\rightarrow V$ be a linear map.
Suppose $U$ is a T-invariant subspace. Define $\overline{T}:V/U \rightarrow V/U$ by $\overline{T}(v+U)=T(v)+U$.
...
- 303
0
votes
1
answer
90
views
I cannot prove that $T$ is diagonalizable $\Rightarrow$ $T|_{V_1},T|_{V_2},\dots,T|_{V_r}$ are diagonalizable.
Let $V$ be a vector space over $\mathbb{F}$.
Let $T\in\mathcal{L}(V)$.
Suppose that $T$ has a matrix $B=\begin{pmatrix}B_1&&&\\&B_2&&\\&&\ddots&\\&&&B_r\...
- 8,410
1
vote
1
answer
92
views
Finding $2$-dimensional invariant subspace
I just want to check that my understanding is correct of invariant subspaces. I was given a matrix A in which I have found that it is invertible, so I know that a $2$-dimensional invariant subspace ...
- 33