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 $~T~$.

Filter by
Sorted by
Tagged with
0
votes
0answers
15 views

Primary Decomposition Theorem examples

The primary decomposition theorem: Let $T:V→V$ be linear operator whose minimal polynomial factors into monic, irreducible, pairwise coprime polynomials: $m(t)=p_1 (t)\cdots p_k (t)$. Then: $V$ ...
0
votes
1answer
39 views

Theorem about Invariant subspace and Mapping Restriction

Can someone provide a proof for the following theorem and explain why $R$ is exactly the definition restricting a linear mapping (operator) like $A$$V$ --> $V$, where $V$ is a finite-dimensional ...
6
votes
2answers
85 views

Proving range is largest subspace of projection operator

Suppose $P: V \rightarrow V$ be a linear map transformation such that $P^{2}=P .$ I have to describe the largest subspace which is pointwise invariant under $P$. I think the largest subspace should be ...
2
votes
1answer
55 views

A invertible iff both restrictions are invertible

I am trying to solve the problem below but without luck: Let V be a finite-dimensional vector space, A $\in $End(V), and U $\subseteq$ V an invariant subspace. Let $A_r \in End(U)$ denote the ...
0
votes
1answer
11 views

Subspaces Question

enter image description here The following happen to all be true. But why is this closed under scalar multiplication? α[[a],[3a+b],[4a+4b],[4a+3b]] =[[aα],[3aα+bα],[4aα+4bα],[4aα+3bα]] and sure, this ...
1
vote
0answers
15 views

Invariant set for the heat equation

I have problems proving that a set of temperature distributions is invariant. I've been looking a lot for material related to my problem, but I was unable to find the correct keywords or relate the ...
0
votes
1answer
34 views

Do the rows of A form a basis for $R^5$?

Suppose $ \begin{vmatrix} 1 & 1 & 0 & 1 & 0 \\ 2 & 1 & 0 & 1 & 0 \\ 1 & 2 & 0 & 1 & 1 \\ 1 & 1 & 0 & 2 & 1 \\ 0 &...
1
vote
1answer
39 views

Which vectors in $S_3$ belong to $span(S_1)$

$ S_3 = \{w1, w2, w3, w4\}, w1 = \left(\begin{array}\ 2 \\ 1 \\ 0 \\ 2 \\ \end{array}\right), w2 = \left(\begin{array}\ 1 \\ 0 \\ 2 \\ 2 \\ \end{array}\right), w3 = \left(\begin{array}\ 0 \\ ...
0
votes
1answer
65 views

Which of the following subsets of $\mathbb{R}^3$ are subspaces of $\mathbb{R}^3$?

(1) A = { (a,b,2a-3b) | a,b $\in $ R } $ \quad$ $A = span\{ (1,0,2) , (0,1,-3) \}\ {\rm is\ a\ subspace\ of}\ \mathbb{R}^3$ (2) $B = \{ (u,v,w) | vw = 0 $ } **$ \quad$ $ Let B = span(T).\ (1,1,0)...
1
vote
1answer
19 views

Show x(t) is an element of a subspace E for all t [duplicate]

Let T be a linear transformation on $\Bbb R^n$ that leaves a subspace $E\subset \Bbb R^n$ invariant and let T(x) = Ax with respect to the standard basis for $\Bbb R^n$. Show that if x(t) is the ...
0
votes
2answers
36 views

Dimension of Subspace in can $\Bbb{R}^5$

Dimension of Subspace For this question, I understand the minimum dimension of $W$ must be $2$ because of the rank nullity theorem. So $5-3 = 2$. However, Would the dimension of $W$ be at most $4$? I ...
0
votes
1answer
20 views

Finding the dimension of a polynomial subspace

Polynomial Subspace For this question (in the attached link above) I originally thought the answer was 7 because typically the equation for finding the dimension of a polynomial is Pn+1 --> in this ...
1
vote
2answers
27 views

Why we can say that $T(\alpha_{i}+\alpha{j})=\lambda(\alpha_{i}+\alpha_{j})$ in a T-invariant subspace?

My problem have the hypothesis that every subspace of V is T-invariant (with T a lineal operator over V). Then I've to prove that T is a scalar multiply of the identity operator. There are some ...
1
vote
0answers
35 views

How to find invariant-subspaces in general?

I noticed there are some similar problems like this in the forum, but actually none of them answer my question in a satisfied way, I think it's still needed to ask a new question on this. I first met ...
1
vote
1answer
74 views

How do I find all the subspaces of $\mathbb{F}^n$ that are invariant under $\operatorname{GL}_n(\mathbb{F})$?

Let $n \in \mathbb{N}$ and $\mathbb{F}$ be a field. The set of all invertible $n \times n$ matrices over $\mathbb{F}$ is denoted by $\operatorname{GL}_n(\mathbb{F})$. A subspace $W$ of $\mathbb{F}^n$ ...
0
votes
1answer
45 views

Cover of set in $\mathbb{R}$ with linear functions

Let $f_1,...,f_N$ be non-degenerate linear functions on $\mathbb{R} \to \mathbb{R}$. Is it possible to create an infinite and well ordered set $S \subset \mathbb{R}$ so that $$f_i(S) \subset \bigcup_{...
0
votes
1answer
35 views

Let $T$ be a linear operator on $V$.If every subspace of $V$ is invariant under $T$,then $T$ is a scalar multiple of the identity operator.

Let $T$ be a linear operator on $V$.If every subspace of $V$ is invariant under $T$,then $T$ is a scalar multiple of the identity operator. This problem is from Hoffman Kunze form chapter Invariant ...
1
vote
0answers
20 views

Why are rational functions invariant under symmetric linear group generated by Plucker coordinates

I am working on an exercise and not sure where to start. Let $K=(\mathbb{C}^2)^{n+3}$. The special linear group $SL_2$ acts naturally on each $\mathbb{C}^2$ and hence on $K$. Let $R$ be the field of ...
0
votes
0answers
49 views

Irreducible Representation of Lie algebra

Let $g$ be a lie algebra, and $V$ be a vector space over $F.$ Assume that $\operatorname{char} (F) =0,$ and $\dim(V)=n>1.$ I want to show that $(\rho, V)$ be in irreducible representation of $g$ ...
1
vote
1answer
15 views

Invariant subspaces of derivative transformation and integral transformation on the linear space of polynomial ring.

$\mathbb{F}$ is a field,$V=\mathbb{F}[x]$,$D$ and $S$ are derivative transformation and integral transformation on$V$. $$D:p(x)\mapsto p'(x).$$ $$S:p(x)\mapsto \int_{0}^{x}p(t)\mathrm{d}t.$$ I want to ...
0
votes
1answer
36 views

Can every vector space,V be written as direct sum decomposition of its proper subspaces which are T-invariant given T is a linear operator from V to V

Since there are linear operators which have no eigen values, that's why this question rose in my mind as whether every vector space can be written as direct sum decomposition of T-invariant PROPER ...
0
votes
0answers
19 views

Is the Matrix subset a Subspace?

If 𝑀2×2 is the vector space of 2×2 real matrices. Let 𝑊 be the set of all 2×2 matrices whose two columns are orthogonal to one another. I believe it's(W) no it is not a subspace for the 2 ...
1
vote
1answer
24 views

an affine transformation that maps an affine subspace on a parallel subspace is a dilation

I thought that an affine transformation that maps an affine subspace on a parallel subspace is a dilation and in the other direction. I tried to prove this. So first I assumed that $F$ is an affine ...
0
votes
1answer
65 views

Invariant subspace of differentiation operator

Let $D \in \mathcal{L}(\mathcal{P}(\mathbb{R}))$ be the differentiation operator, and let $U$ be an invariant subspace of $D$. Suppose there exists a $p \in U$ with deg $p = k$. a) Show that $\mathcal{...
4
votes
2answers
105 views

Characterization for invariant subspace

I have the following problem consisting of 3 parts of which I'm not being able to figure out the last. Notation: $T^*$ is the adjoint operator of $T$. $Im(T) = \{T(v) : v \in V\}$. $Ker(T) = \{v \in V ...
-1
votes
3answers
57 views

Why does $(Av)\cdot (Aw)=v\cdot w$ hold?

Let $A\in M_n(\mathbb{R})$ be orthogonal and $U\leq_A V$, where $V=\mathbb{R}^n$ Then we have that $U$ $\ A$-invariant, this means that $Au \in U$ for all $u \in U$. How can we show that $(Av)\cdot (...
2
votes
0answers
76 views

Toeplitz Kernel Question

$$\newcommand{\kern}{\operatorname{Ker}}$$When looking at a Toeplitz Operator (on $\mathbb{H}^2(\mathbb{C^+}))$ with unimodular symbol, $\phi=e^{i \psi}$, what can we say about $\kern T_{\phi}$ when $$...
1
vote
1answer
46 views

Hoffman Kunze problem 12 section6.4

Let $T$ be a linear operator on a finite dimensional vector space over an algebraically closed field $\Bbb{F}$. Let $f$ be a polynomial over $\Bbb{F}$. Prove that $c$ is a characteristic value of $f(T)...
1
vote
1answer
28 views

A problem on invariant subspaces.

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. Attempt Suppose $f$ ...
0
votes
1answer
49 views

Show that the subspace $U$ is $\phi^z$-invariant

Let $\mathbb{K}$ be a field and let $V$ be a $\mathbb{K}$-vector space. Let $\phi,\psi:V\rightarrow V$ be linear maps, such that $\phi\circ\psi=\psi\circ\phi$. I have shown using induction that if $U\...
2
votes
1answer
33 views

$W$ is a $T$-invariant subspace of $V$, prove a Jordan form of $T|_W$ contained the Jordan form of $T$.

The meaning of the title is to show that each block of the Jordan form of $T|_W$ corresponds to a block in the Jordan form of $T$ of equal or greater size. I know that each Jordan block in the form of ...
1
vote
1answer
46 views

$Z(v,T) = 1 \iff v$ is engevector of $T$.

$\textbf{Definition:}$ Let $V$ be a finite dimensional vector space over a field $F$ and let $T:V \to V$ be a linear operator. If $v$ is a vector in $V$, the $T-$cyclic subspace generated by $v$ is ...
0
votes
1answer
22 views

Intersection between invariant curves for a map that coming from an autonomus vector field.

Consider the system of differential equation $$\dot x = f(x) $$ with $x\in{M}$ where $f$ is a function of the only $x$, so the system is autonomus. Now let $\phi^t(x)$ be the flow of the system, that ...
0
votes
1answer
25 views

Question about $T-$cyclic subspace generated

$\textbf{Definition:}$ Let $V$ be a finite dimensional vector space over a field $F$ and let $T:V \to V$ be a linear operator. If $v$ is a vector in $V$, the $T-$cyclic subspace generated by $v$ is ...
3
votes
1answer
42 views

$T$ is a normal operator, prove any eigenspace of $T+T^*$ is invariant under $T$

I've been asked to prove in a homework problem exactly what the title describes. This was the $3^{rd}$ part of a question whose first 2 parts were to prove that $\ker T=\ker TT^*$ and $\ker T=\ker T^n$...
0
votes
0answers
24 views

Reducible/Irreducible 2x2 matrices

as far as I understand, one considers that an irreducible matrix only contains trivial invariant subspaces (the whole space, the null vector and its eigenvectors). So, are there 2x2 reducible matrices?...
1
vote
0answers
43 views

Invariant subspaces of symmetric operators

I haven't found anywhere that discusses representations of the symmetric group in a way that applies to this situation, and I'd appreciate any pointers/resources/phrases-to-Google. Consider a vector ...
2
votes
1answer
29 views

How do I prove that eigenspaces and root subspaces are invariant for A?

So the eigenspace is $Ker(A-λI)$ where $λ$ is an eigenvalue of A and the root subspace is $Ker(A-λI)^r$ where $r$ is the exponent of $(x-λ)$ in the minimal polynomial for $A$. My professor stated that ...
1
vote
1answer
35 views

Show that $\phi (v)=\lambda v$ for a vector $v$ and a coefficient $\lambda$

Let $\mathbb{K}$ be a field, $1\leq n\in \mathbb{N}$ and let $V$ be a $\mathbb{K}$-vector space with $\dim_{\mathbb{R}}V=n$. Let $\phi :V\rightarrow V$ be a linear map. The following two statements ...
2
votes
1answer
39 views

$M_B(\phi)$ is an upper triangular matrix iff $U_i\subset U_{i+1}$ and $U_i$ is $\phi$-invariant

Let $\mathbb{K}$ be a field, $1\leq n\in \mathbb{N}$ and let $V$ be a $\mathbb{K}$-vector space with $\dim_{\mathbb{R}}V=n$. Let $\phi :V\rightarrow V$ be a linear map. I want to show that the ...
0
votes
1answer
112 views

Eigenvalues of a linear matrix pencil (Ax = λ Bx) using the subspace iteration method

I am trying to find the rightmost eigenvalues of the generalized eigenvalue problem ($Ax = \lambda Bx$) using the subspace iteration method. This formulation arises from flow stability analysis where ...
0
votes
1answer
36 views

Cyclical subspace intuition

can someone give an intuition for what Cyclical subspace means ? also we saw the given a non zero vector x, a basis for Cyclical subspace is: $$W = span({x,T(x),T^2(x),...})$$ can you explain how this ...
0
votes
1answer
48 views

Intersection of two invariant subspaces

I need ro find all subspaces of $\mathbb{R}^3$ that are invariant simultaneously with respect to two linear transformations defined by matrices $$ A=\begin{pmatrix} 5& -1& -1\\ -1& 5&...
0
votes
1answer
24 views

Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $T|_W$ for any $T$-invariant $W.$

Let $V$ be a complex vector space with a linear operator $T : V \to V$ and a $T$-invariant subspace $W \subseteq V.$ Prove that the Jordan Canonical Form of $T$ contains the Jordan Canonical Form of $...
1
vote
2answers
42 views

Orthogonal tranformation over invariant subspace [duplicate]

let $V$ be a euclidean space. $T:V\to V$ be an orthogonal linear transformation. $W\subset V$ is a $T$-invariant subspace. I need to prove 2 things: A. $T\bigl|_W$ is orthogonal so I said that if ...
2
votes
0answers
41 views

Application of bounded operators [closed]

By definition, a bounded linear operator $T: X \to X$ has a non-trivial closed invariant subspace W on a Banach space $X$ if for every vector $\omega \in W$, $T(\omega)$ belongs to $W$ (i.e, $T(W) \...
0
votes
0answers
25 views

Decomposition of vector space to produce nilpotent and invertible transformations [duplicate]

Let $V$ be a finite-dimensional vector space over a field $K$. Let $T$ be a linear operator on $V$. Prove that there exists a unique sum $V=V_{0}+V_{1}$ such $T(V_{0}) \subseteq V_{0}$, $T(V_{1}) \...
1
vote
0answers
24 views

Spectral properties of hypercyclic operators

I'm studying some topics related to the invariant subspace problem, and consequently I find myself dealing with hypercyclic operators. (An operator $T:X\rightarrow X$ is hypercyclic if there is some ...
0
votes
1answer
41 views

Find the equation of a plane that is invariant with respect to the following transformation:

Find the equation of a plane that is invariant with respect to the following transformation: \begin{pmatrix}4&-23&17\\ \:11&-43&30\\ \:15&-54&37\end{pmatrix} Actually, I don'...
0
votes
2answers
19 views

Linear map triangulizable

What's the definition of a linear map that is triangulazible? I can't find it anywhere. In addition, I was asked to find a linear map that doesn't have any invariant sub-spaces. I know that if a map ...

1
2 3 4 5 6