# 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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### 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 ...
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### 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 $v$...
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### 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 ...
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### 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 ...
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### 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 $C^{2}\otimes C^{2}\otimes C^{2}=2C^2\bigoplus C^4$...
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### 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 ...
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$T: W \rightarrow W$ 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 $$W=W_1\... 0 votes 0 answers 15 views ### Show that there exists exactly one l_A-invariant K-subspace of dimension 1 in K^2. So I have this exercise where I want to check my solutions. Can someone help me? Let A = \begin{pmatrix} a &b \\ 0& d \end{pmatrix} ∈ Mat_{2,2}(K) with a, b, d ∈ K, where a ≠ 0 and d ≠ ... 0 votes 0 answers 30 views ### Boundary conditions and invariant sets for PDE's I'm looking into invariance set theorems for parabolic PDE's (such as https://memoriam.cse.umn.edu/hfw/Math/InvariantSets.pdf and https://arxiv.org/pdf/0911.4526.pdf). However, they don't specify ... 3 votes 1 answer 51 views ### Spanning set of symmetric invariants of tensor powers Let M be a module over a commutative ring R and let the symmetric group \Sigma_n act on M^n and M^{\otimes n} by permuting factors. For v \in M^n, let \text{Stab}(v) denote the ... 0 votes 0 answers 45 views ### Exercise 4, Section 6.7 of Hoffman’s Linear Algebra Let T be a linear operator on V. Suppose V=W_1\oplus … \oplus W_k,where each W_i is invariant under T. Let T_i be the induced (restriction) operator on W_i. (a) Prove that \det (T)=\det ... 1 vote 0 answers 46 views ### Exercise 1, Section 6.7 of Hoffman’s Linear Algebra Let E be a projection of V and let T be a linear operator on V. Prove that the range of E is invariant under T if and only if ETE=TE. Prove that both the range and null space of E are ... 0 votes 0 answers 63 views ### Theorem 10, Section 6.7 of Hoffman’s Linear Algebra Theorem 9: V=W_1\oplus …\oplus W_k$$\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. ... 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-... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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_{\... 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 ... 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(... 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 ... 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) \... 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 ... 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 ... 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{...
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 $\overline{T}(v+U)=T(v)+U$. ...