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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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{...
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Construction of invariant subspaces

Let $V$ be a finite dimensional real vector space and $f$ an endomorphism on $V$. Show that there exists a subspace $0\neq U \subseteq V$ such that $f(U) \subseteq U$ $\dim(U) \leq2.$ Unless I'm ...
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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 $\overline{T}(v+U)=T(v)+U$. ...
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block matrix of an invariant subspace

Let $V$ be a vector space over the field $K$ and $\dim V = n$. Furthermore, let $U$ be a subspace of $V$ with $\phi(U)\subset U$. $v_1,..,v_k$ is a basis of $U$ and we can complete the basis with $v_{...
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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\...
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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 ...
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What do you call the converse of an invariant subspace of an operator?

Question. I am looking for the concept converse to invariance: what do we call a set $W$, such that $$T(w) \in W \implies w \in W ?\tag{1}$$ I feel there was a word for this, but I can't recall it, ...
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Need help for understanding a sum of subspaces

I am newbie started learning the linear algebra. It might be dumb question. But I don't understand how the sum of subspace can also be subspace?! So for subset in order to be subspace, It should ...
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2 answers
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How to find all the invariant subspaces in relation to Matrix

Let $$A=\begin{bmatrix}1 & 1 & 0 & 0\\ -1 & 1 & 2 & 1\\ 0 & 0 & 3 & 1\\ 0 & 0 & -1 & 1 \end{bmatrix}\in M_4(\mathbb C)$$ I need to find all the $A$-...
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Finding a invariant subspaces for a specific matrix, but this time: over C

The matrix I found a good algorithm here, ( Finding a invariant subspaces for a specific matrix ) but this time I want to find the invariant subspaces for a specific matrix OVER C (complex). does it ...
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Invariant and cyclic subspaces

Iv'e been given this question and I don't really have a clue how to do it. Let $V$ be a finite dimensional vector space of dimension $n$ over a field $\mathbb{F}$, and let $f$ be an operator over $V$. ...
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Is the assumption that $V$ is finite-dimensional really necessary in Exercise 5.A.28? (Sheldon Axler "Linear Algebra Done Right 3rd Edition")

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. The author assumed that $V$ is finite-dimensional in Exercise 5.A.28. But I don't think that this assumption is ...
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Exercise 5.A.10(b) Find all invariant subspaces of $T$ ("Linear Algebra Done Right 3rd Edition" by Sheldon Axler)

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. The following exercise is Exercise 5.A.10 on p.139. Define $T\in\mathcal{L}(\mathbb{F}^n)$ by $$T(x_1,x_2,x_3,\dots,...
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The maximal number of orthogonal projections with images invariant under each other

Let $V=\mathbb{R}^n$ and $Gr_k(V)$ be the set of all $k$-dimensional subspaces of $V$. For $W\in V$, denote $P_W: V\to W$ the orthogonal projection onto $W$. Suppose $\mathcal{S}\subset Gr_k(V)$ ...
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Determining whether a set of vectors is a subspace (given an equation with matrices?) [closed]

So I have this question that I just don't know how to set up for my linear algebra class. It involves the concept of subspaces which is something I've struggled on in this class for sure, it would be ...
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Invariant subspaces concept - linear algebra

Why are invariant subspaces called invariant? Normally invariant in mathematics means something that doesn't change. Suppose we have a linear map $T:V \to V$ Also suppose $P$ is a proper subspace of $...
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How to write a subset of $\mathbb{C}^4$ as a subset of $\mathbb{C}^3$

I'm currently working with the standard representation of the symmetric group $S_n$. Recall that the standard representation of $S_n$ permutes the basis elements of the $n \times n$ identity matrix ...
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How to find invariant subspace?

If I have a multiplication operator $L_A: \mathbb{R}^4 \to \mathbb{R}^4$, where $$ A=\begin{pmatrix} 43 & -32 & 35 & 14 \\ -66 & 44 & -52 & -18 \\ -137 & 97 & -110 &...
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A problem about cyclic subspaces and minimal polynomial

Let $\alpha$ be a linear operator on a vector space $V$, and supoose that $V$ is $\alpha$-cyclic, say generated by $v\in V$. Suppose further that $V=U_1\bigoplus U_2$ for non-trivial $\alpha$-...
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An application of Zorn's Lemma in linear algebra

Let $\alpha$ be a linear operator on a vector space $V$. Let $v_0\in V$ and suppose that the minimal polynomials of $\alpha$ is $m_\alpha(x)$ and of $\alpha$ at $v_0$ is $m_{\alpha,\ v_0}(x)$. Suppose ...
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Is $ k_1a + k_2b + k_3c = 0$ for arbitrary but fixed numbers $ k_1, k_2, k_3 \in \mathbb{R} $ a subspace in $\mathbb{R^3}$? [closed]

Need some guidance at this task: I have $k_1a + k_2b + k_3c = 0$ for arbitrary but fixed numbers $k_1, k_2, k_3 \in \mathbb{R}$ and need to show if (let's call it $V$) is a subspace of $\mathbb{R^3}$. ...
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Unitary transformation with T invariant subspace

I encountered the following problem and I was wondering whether it is possible to deduce even stronger conclusion that $T = \pm I$ This is the problem: Let $T: V \to V$ be a linear transformation on ...
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Proving that the smallest dimension for $V$ is 4.

Let $V$ a $\mathbb{Q}$-linear space, $\dim_\mathbb{Q}V<\infty$, $T: V \rightarrow V$ a linear operator such that $T^2 = -Id$. If $V$ has a $T$-invariant proper subspace $W$, $\dim(W) \ge 1$, then ...
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If the subspace $W$ is $T$-invariant, then $W^\perp$ is $T^*$-invariant.

Let $V$ be complex inner product space and let $T$ be mapping from $V$ to $V$. Prove that if the subspace $W$ is $T$-invariant, then $W^\perp$ is $T^*$-invariant. I would like to ask if my proof is ...
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Examine the eigenvalues of a linear map that has only two invariant subspaces

Let V be a vector space (over $\mathbb{F}=\mathbb{R}$ of $\mathbb{C}$) of dimension $n\geqslant 2$, such that $f$ has only $\{0_V\}$ and $V$ as it's invariant subspaces. Examine if $f$ has eigenvalues....
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Define the smallest subspace of $3\times 3$ matrix vector space that contains the set of all invertible matricies.

Define the smallest subspace of $3\times 3$ matrix vector space that contains the set of all invertible matrices. My attempt: $A$ is invertible if and only if its columns form a basis in $F^3$. The ...
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A matrix $A$ is $K$-irreducible if and only if no eigenvector lies on $\partial K$.

I'm studying the properties of nonnegative matrices, and I encountered a theorem for which I can not understand its proof. The theorem can be found in "Nonnegative matrices in the Mathematical ...
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The plane described by the equation $ax+by+cz+d=0$ is invariant under the affine map $f$ if and only if $(a,b,c,d)$ is an eigenvector of $M$.

My lecture notes include the following result with no proof: Theorem: Let $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be an affine map with corresponding linear function $\phi:\mathbb{R}^3\rightarrow\...
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Does S form a basis for R2?

I have a question here i can not explain my answer if it is right i do not sure Let S = {A = (a1, a2) , B = (a2, b2)} be a spanning set for R2 and some element x = (a, b) ∈ R2 We have x = c1 A + c2 B ...
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5 votes
1 answer
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Invariant subspace problem for $\ell^2(\mathbb{N})$

Is the invariant subspace problem known for $\ell^2(\mathbb{N})$ or for more general $L^2$ spaces, i.e. does every bounded linear operator $T \colon \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N})$ have a ...
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1 vote
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Equivalence of different definitions of cyclic subspace and invariant subspace.

I just read about the notes of functional analysis from my teacher. It gives a definition on cyclic subspace and invariant subspace as following: Let H be a Hilbert space, L is a closed subspace, T is ...
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4 votes
2 answers
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Invariant Subsapce of Real vector space

Let $V$ be a real vector space of dimension at least $3$ and let $T\in \operatorname{End}_{\mathbb{R}}(V)$. Prove that there is a non-zero subspace $W$, $W\neq V$, such that $T(W)\subseteq W$. I can ...
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Matrices and Linear Algebra [closed]

Let A be a real $n × n$ orthogonal matrix. Let $X$ be a complex eigenvector of A with complex eigenvalue λ. Prove that $X^TX$ = $0$. Write $X = R + Si$ where R and S are real vectors. Prove that W ...
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Invariant subspaces and eigenvalues

During a lecture my professor explained, that if $A\in \mathbb C^{n\times n},B\in \mathbb C^{k\times k}$ and $C\in\mathbb C^{n\times k}$ with $1\le \operatorname{rank}(C)=k\le n\ $ such that $AC=CB \ $...
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Given a linear map $T : \mathbb{Q}^4 \rightarrow \mathbb{Q}^4$, there exists a nonzero proper subspace $V$ such that $T(V ) \subset V$.

Given a linear transformation $T : \mathbb{Q}^4 \rightarrow \mathbb{Q}^4$, there exists a nonzero proper subspace $V$ of $\mathbb{Q}^4$ such that $T(V ) \subseteq V$. Is the statement true? I think ...
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1 vote
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Determine all $T$-invariant subspaces

Exercise Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be the linear operator given by $$T\begin{pmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{pmatrix} = \begin{bmatrix} x_3 \\ x_1 \\ x_2 \...
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3 votes
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Prove that if $U$ and $W$ are subsaces of $V$ with $V = U+W$ then there exists a subspace $W_1$ of $W$so that $V=U \oplus W_1$

Prove that if $U$ and $W$ are subsaces of $V$ with $V = U+W$ then there exists a subspace $W_1$ of $W$so that $V=U \oplus W_1$ My attempt: Let $W_1 = \big\{w \in W | w \notin W \cap U \big\}$. Then $...
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Invariant subspace related problem

$ \mathrel y \in C^n and y\neq 0 $ and m be the smallest integer such that $ \{y,By,...,B^my\} $ is a dependent set. How to prove that $ V = span \{y,By,...,B^{m-1}y\} $ is B−invariant. I came across ...
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I was reading a linear algebra book when I came across a statement that confused me. Help me if I misunderstood anything.

A book that I'm reading stated "let $V$ be a vector space and let $D$ be a nonempty subset of $V$. Let $M$ be the collection of all vectors in $V$ which can be expressed as a linear combination ...
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$T$-admissible subspace: an equivalent way?

In the book of Linear Algebra by Hoffman-Kunze, the authors write Let $T$ be a linear operator on a vector space $V$ and let $W$ be a subspace of $V$. We say that $W$ is $T$-admissible if $W$ is ...
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Invariant space of a matrix

Assume that $A$ is matrix with a unique complex simple (up to conjugation) leading eigenvalue of length $1$. Let $\lambda$ be the complex eigenvalue $|\lambda|=1$ and let $v=x+iy$ be a complex ...
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Properties of an endomorphism and its minimal polinomial

Recently, I have found this statement and I try to solve as exercise. Consider a vector space $V$ with finte dimension $n\geq 1$ and $T\in End(V)$ with $m^T(t)=\prod_{i=1}^{r}p_i(t)^{e_i}$, where $p_i(...
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For a real symmetric matrix $A$, are the subspaces given by the span of eigenvectors the only $A$-invariant subspaces?

Given a real symmetric matrix $A$, is it true that any $A$-invariant subspace is formed by the linear span of some subset of eigenvectors of $A$?
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Subspaces in linear algebra - Matrix

Let $S=$ {$a_{ij} \in M_{3}(\mathbb{R}):a_{11}+a_{12}+a_{13}=a_{21}+a_{22}+a_{23}=a_{31}+a_{32}+a_{33}$} $S$ is a subspace of $M_{3}(\mathbb{R})$ and dim $S = 7$ I tought I could arrive somewhere ...
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2 answers
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Eigenvalues-eigenvectors

Let $A \in \mathbb{R}^{3\times3}$ be a matrix with the eigenvalues $\lambda_1 = -1, \lambda_2 = 1$ and $\lambda_3 = 5$. Show that there is a 2-dimensional subspace $U \subset \mathbb{R}^{3}$ such ...
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Invariant subspace of linear mapping

Let $T$ be a linear transformation on a complex vector space $V$ of dimension $4$ and let $\lambda_1,\lambda_2,\lambda_3$ be distinct eigenvalues of $T$. Eigenvectors are: $\lambda_1\rightarrow \{v_1,...
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2 votes
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Proof that any every operator on a finite-dimensional,nonzero,real vector space has an invariant subspace of dimension 1 or 2

in Steven Roman's Advanced linear algebra, the author prove the theorem in the following way: suppose that f is a real linear operator and then factors its minimal polynomial m(x) into a product of ...
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Explanation for Partial Diagonalization of a Matrix

Just to note, I haven't yet studied Jordan Normal Form or similar. I am working through a question that contains the following: Say we have $$A=\begin{pmatrix} 0&1&0\\ 0&0&1\\ 1&0&...
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$T: V \rightarrow V$ is a linear operator, and $TT^*=T^3$, $\text{Ker}(T) = U, W=U^⊥$, Prove that $T_{|_W}$ is normal

I proved that $W$ is $T^*-\text{invariant}$, and that $W$ is $T-\text{invariant}$. I couldn't find a good way to prove that the operator $T_{|_W}$ is normal. here's what I did so far: $U=\text{Ker}(T)...
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Expressing transformations as polynomials of $T$ over $T$-cyclic space [duplicate]

I'm reading a book about linear algebra (Lineare Algebra, Bröcker) and it contains the following exercise: Let $T \in End_{K}(V)$ be nilpotent and V be a $T$-cyclic vector space ($V = \langle x,Tx,T^...
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