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Questions tagged [invariant-subspace]

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1answer
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How to see if a subspace is contained in another subspace?

So basically I have a problem with 2 subspaces given in the following spans $$U=\mathscr L\{(1,2,-1,3),(2,4,1,-2),(3,6,3,-7)\}$$$$V=\mathscr L\{(1,2,-4,11),(2,4,0,14)\}$$ And I am asked if it is true ...
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3answers
61 views

On Minimal polynomial and Invariant subspace

Given $T:V \rightarrow V$, $V$ is a vector space over field $\mathbb{R}$ and $m_T = (x^2-2x+2)(x-3)^2$. Show that there exists an invariant subspace with dimension $2$. I first thought that since $3$ ...
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2answers
28 views

Invariant Subspaces and Orthogonality

I'm struggling with a question on invariant subspaces. If someone could possibly help me out here that would be great. The question is as follows: Let $A ∈ M$ where $M$ is the set of $n \times n$ ...
3
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1answer
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Does every eigenspace of the exterior power $\bigwedge^k A$ corresponds to an invariant subspace?

Let $V$ be an $n$-dimensional real vector space, and let $1<k<n$ be fixed. Given an automorphism $A \in \text{GL}(V)$, consider its $k$-th exterior power $\bigwedge^k A \in \text{GL}(V)$. ...
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1answer
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An example for an invertible matrix with no $3$-dimensional invariant subspaces

Is there an example for an invertible linear transformation $\mathbb R^n \to \mathbb R^n$ with no $3$-dimensional invariant subspaces (where $n \ge 4$)? Every real linear transformation admits $2$-...
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2answers
69 views

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 ...
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0answers
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Is an invariant subspace always spanned by a set of generalized eigenvectors?

I have been trying to find an answer to this question for some time and haven't had any luck. Let me state the question formally: Suppose $T$ is a linear mapping $T:V \rightarrow V$, where $V$ is an $...
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2answers
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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 ...
2
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3answers
58 views

What if there are two non-orthogonal invariant subspaces?

Let $U_1,U_2\subseteq\Bbb R^n$ be two invariant subspaces w.r.t. some group $\Gamma\subseteq\mathrm O(\Bbb R^n)$ of orthogonal matrices. I wonder the following: Question: If $U_1$ and $U_2$ are ...
2
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1answer
49 views

Finding the submodules of the $\mathbb{R}[x]$-module defined by $A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & -1 \end{bmatrix} $

I'm trying to solve a question which asks me to consider the matrix $A$ with $$A=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 &0 & -1 \end{bmatrix} $$ ...
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1answer
48 views

Prove that the image of an invariant subspace under a morphism of representations is an invariant subspace.

Prove that the image of an invariant subspace under a morphism of representations is an invariant subspace. Could anyone give me a hint on how to solve this Please? Knowing the following: And ...
0
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1answer
20 views

Why is $\mathcal{K}$ invariant under $A$?

I'm reading Halmos' Finite Dimensional Vector Spaces. Here: I am a little bit confused at to why $\mathcal{K}$ is invariant under $A$. If $k\in \mathcal{K}$, why does it follows that $Ak\in ...
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0answers
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Finding all subspaces invariant under F.(2)

I have a question regarding the question in the link here: Finding all subspaces invariant under F. The answer is so convincing in the above link but I just want to know why the matrix corresponding ...
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1answer
44 views

Finding all subspaces invariant under F.

The question and its answer are given below: But I do not know why is this the answer, could anyone explain this for me please?
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1answer
100 views

Is this representation completely reducible?

Is these representations completely reducible? Definition: A linear representation is said to be completely reducible if every invariant subspace has an invariant complement. But I have no idea how ...
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1answer
49 views

Let $v_1$, $v_2$, $v_3$ be a basis of the $\mathbb{R}$-vector space $\mathbb{R}^3$

I'm not that good at math and would be very happy if you could give me some hints and so on So my task is: Let $\{v_1, v_2, v_3\}$ be a basis of the $\mathbb{R}$-vector space $\mathbb{R}^3$ show ...
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37 views

Decomposition of invariant subspaces

Consider a set of $d$ linearly independent generalized eigenvectors of some matrix $A \in \mathbb{C}^{d \times d}$. Suppose this set is decomposed into $M$ distinct Jordan chains and the $m$-th Jordan ...
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Application of the cyclic decomposition theorem to a linear transformation

Given a linear transformation $L:V→V$ where $V$ has basis $\{v_1, v_2, v_3\}$ with $L(v_i) = v_i$ for all $i < 3$ and $L(v_3) = v_1 + v_2$, decompose $V$ as a direct sum of $T$-cyclic subspaces as ...
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0answers
28 views

Invariant subspace concept - generalization

Suppose $A: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $B: \mathbb{R}^n \rightarrow \mathbb{R}^m$ are linear maps. Let $X \subset \mathbb{R}^n$ be such a linear subspace of $\mathbb{R}^n$ that $A(X)=B(...
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1answer
46 views

$T$ and $U$ have a common eigenvector, where $T$ and $U$ are linear operators on odd dimensional vector space $V$ and $T^2 = U^2 = I$.

I am trying to prove that $T$ and $U$ have a common eigenvector, where $T$ and $U$ are linear operators on odd dimensional vector space $V$ and $T^2 = U^2 = I$. I have been stuck on this problem for ...
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1answer
72 views

How to represent characteristic polynomial in terms of those of invariant direct sum subspaces?

Suppose $V$ is a complex vector space and $V_1,...,V_m$ are nonzero subspaces of $V$ such that $V = V_1 \oplus ... \oplus V_m$. Suppose $T \in \mathcal{L}(V)$ and each $V_j$ is invariant under $T$. ...
2
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1answer
54 views

Is the set of all vectors in the span of $\pmatrix{1 \\ 2}$ and $\pmatrix{2 \\ -1}$ a subspace or not?

I'm trying to figure out whether or not the set of all vectors in the span of $\pmatrix{1 \\ 2}$ and $\pmatrix{2 \\ -1}$ is a subspace or not (I know the answer to be yes, it is a subspace but I want ...
0
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1answer
20 views

Prove that if $\phi$ is invertible operator of V space, then $\phi$ and $\phi^{-1}$ has the same invariant subspaces.

Prove that if $\phi$ is invertible operator of V space, then $\phi$ and $\phi^{-1}$ has the same invariant subspaces. I know how to prove this statement for finite V. (Wish it is right) : Let $W$ be ...
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1answer
27 views

Invariant subspace of two operators T and U when $TU=UT$

$V$ is a finite dimensional vector space. Suppose I have tow linear operators $T,U$ on $V$ such that $TU=UT$. We know that the range of $T$ and $\ker T$ is a invariant subspace of $U$. This motivates ...
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0answers
9 views

Irreducibility of differential operator?

Is the differential operator $D:P_n \to P_n$ is reducible? Find an element of $P_n$ that is of period $n+1$ under $D$. Here $D$ is Differential operator and $P_n$ is the vector space of all ...
0
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0answers
45 views

Let $B={\{x^2, x, 1}\}$ and $S= {\{x^2+x, 2x-1, x+1}\}$ be two basis of $P_2$.

Let $B={\{x^2, x, 1}\}$ and $S= {\{x^2+x, 2x-1, x+1}\}$ be two basis of $P_2$. Let $i_b$ and $i_s$ be the coordinates maps induced on $P_2$ by these two basis. Let$T: P_2\to P_2$ be a liner ...
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1answer
48 views

Does transformation invariance of the range and null space imply commutativity?

Suppose $R(U)$ and $N(U)$ (the range and null space of a linear transformation $U$, respectively) are $T$-invariant ($T$ linear) subspaces of some vector space $V$. Does this imply $UT=TU$? I've ...
0
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1answer
35 views

Nilpotent Transformations and Invariant Subspaces

I'm trying to work my way through a question I've been set, which is as follows: (i) For $i$ > 0, let $K_i = KerT^i$. Show that for each i, $K_i$ ⊆$ K_{i+1}$, and hence show that there exists a non-...
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0answers
29 views

Are complements of generalized eigenspaces $T$-invariant?

Let $T$ be a linear operator of a finite-dimensional vector space $V$ over the field $F$. Let's assume that $T$ has an eigenvalue $\lambda$. Let $K_\lambda$ be the generalized eigenspace of $T$ ...
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0answers
21 views

How to get density of a region (subspace) in a vector space?

I have a simple problem, which I think must have an easy solution. I have a vector space say with a 1000 dimensions for each vector. Now, I have a large number of sample vectors from this vector ...
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3answers
53 views

Generalized Eigenspaces Associated to Different Values

Let $V$ be some vector space, and $T \in \mathcal{L}(V)$. If $a \neq b$, then $G(a,T) \cap G(b,T) = \{0\}$, where $G(b,T)$ denotes the generalized eigenspace. I am having a lot of trouble with this ...
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1answer
46 views

summation properties of three subspaces?

I have E, F, G as subspaces of V. Confused as to how to start proving that if E + F = E + G, then F = G. Also the same except with direct sum. Assuming it involves evaluating combined summation but ...
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0answers
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A property of normal transformation

Let $A$ be a normal transformation on an Euclidean space $V$.Then how to show that if $W$ is an invariant sub-space of $A$, then $W^{⊥}$ is also an invariant sub-space of $A$. My try: For any$\alpha\...
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1answer
33 views

If $T\in L(V)$ is diagonalizable then show that $R(T)\cap N(T)=\{0\}$

If $T\in L(V)$ is diagonalizable then show that $R(T)\cap N(T)=\{0\}$ My Attempt: Case 1: $V$ is finite dimensional Subcase 1: $T$ is invertible That means $N(T)=\{0\}$ Directly $R(T)\cap N(...
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3answers
34 views

Another way of arguing existence of eigen value for matrix over Complex Number

I Know that by use of fundamental theorem of algebra and by considering Characteristics polynomial we guarantee the existence of eigenvalues in case of complex field. But Is there is any other simple ...
0
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1answer
47 views

If A is diagonalisable then $T\in L(V)$ given by $T(B)=AB-BA$ is diagonalisable

I wanted to prove that If A is diagonalisable then $T\in L(V)$ given by $T(B)=AB-BA$ is diagonalisable. My Attempt: I know that if minimal polynomial of T has distanct linear factors then we can show ...
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1answer
26 views

Commutativity of Operator in Polynomial

I was studying the Following lemma attached in Image given.enter image description here I know that polynomial is the commutative ring.B ut I am not able to convince my self if we write the product ...
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2answers
94 views

Vector subspace of $\mathbb{C}$ spanned by $n$ complex numbers over the field of rational numbers.

Let $\alpha_{1} \dots \alpha_{n}$ be complex numbers and V = $\{ \sum_{i=1}^{n} a_{i}\alpha_{i} : a_{i} \in \mathbb{Q} \}$ be the vector subspace of $\mathbb{C}$ spanned by them over the field of ...
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Finding a specific $T$-invariant subspace that satisfies 3 criteria

Suppose that $V$ is an infinite-dimensional vector space over $F$, and $T : V \rightarrow V$ is linear. Suppose also that $W$ is a $T$-invariant subspace of $V$ . Show that there is a subspace $U$ ...
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2answers
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Finding which sets are subspaces of R3

https://i.stack.imgur.com/Bpl28.png Hello. I have attached an image of the question I am having trouble with. I thought that it was 1,2 and 6 that were subspaces of $\mathbb R^3$. Here is my working:...
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1answer
119 views

Transformation restriction to commutative matrices

Let $T: \Bbb M_{3x3}(\Bbb R) \rightarrow \Bbb M_{3x3}(\Bbb R)$ be a linear transformation such that $T(B) = AB$ where: $$ A= \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ ...
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3answers
51 views

If $W$ is $T$-invariant space of $V$ and $\left\{0\right\}\ne W\subseteq V$ then $\exists\ w\in W$ such that $w$ is an eigenvector of $T$

Assume $V$ is a finite dimensional vector space over $\mathbb{C}$. and $T:V\to V$ is a linear transformation. Show that if $W$ is $T$-invariant space of $V$ and $\left\{0\right\}\ne W\subseteq V$ ...
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2answers
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Proving existence of common eigenvalue using invariant subspace properties

Let $T: \Bbb R^n \rightarrow \Bbb R^n$ be a linear transformation such that every subspace $U \subseteq V$ of dimension $n-1$ is $T$-invariant. I want to prove that there exists a $\lambda$ such that ...
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2answers
55 views

Prove that $W$ is $g(T)$-invariant for any polynomial $g(t)$.

Let $T$ be a linear operator on a vector space $V$, and let $W$ be a $T$-invariant subspace of $V$. Prove that $W$ is $g(T)$-invariant for any polynomial $g(t).$ is an answer that I found online. I ...
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1answer
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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 $k<n$, with $\mathrm{col}(B)$ (the column space of $B$) not an invariant subspace of $A$. We assume $A\neq0,I_n$. ...
2
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1answer
43 views

Compact operator on invariant subspace (not necessarily closed) is compact

I am looking at a problem in Conway's functional analysis text. It is problem II.5.7, which states: "If $T$ is compact and $\mathscr{M}$ is an invariant subspace for $T$, show that $T|_{\mathscr{M}}$ ...
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1answer
45 views

Can one describe the algebraic multiplicity in terms of generalized eigenspaces and the minimal polynomial?

We defined the algebraic multiplicity of a matrix $A$ with eigenvalue $\lambda$ to be the largest integer $r$ such that $(x-\lambda)^r$ divides the characteristic polynomial of $A$. I would like to ...
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0answers
55 views

Show that there is a k dimensional subspace E such that…

I am working on studying for an exam. I was looking at this old problem and was wondering if someone could help me out. So far I think I can say that since k eigenvalues with non-positive real part ...
1
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1answer
36 views

Invariant Subspaces(projection operator) [closed]

Let $\mathbf P$ be the (hermitian) projection operator onto a subspace $M$. Show that $1 -\mathbf P$ projects onto $M^\bot.$ Hint: You need to show that $\langle m \mid \mathbf P\mid a\rangle = \...
0
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1answer
100 views

Restriction of a diagonalizable linear operator $T$ to any nontrivial $T-$invariant subspace is also diagonalizable using the given lemma.

Lemma Let $T$ be a linear operator on a finite-dimensional vector space $V$, and let $W$ be a $T-$invariant subspace of $ V$. Suppose that $v_1, v_2, . . . , v_k$ are eigenvectors of $T$ ...