# 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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### 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 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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### 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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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&...