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invariant subspace (Definition)

Let $ T: V\rightarrow V$ be a linear transformation of a vector space $ V$. A subspace $ U\subset V$ is called a $ T$-invariant subspace if $ T(u)\in U$ for all $ u\in U$.

If $ U$ is an invariant subspace, then the restriction of $ T$ to $ U$ gives a well defined linear transformation of $ U$. Furthermore, suppose that $ V$ is $ n$-dimensional and that $ v_1,\ldots, v_n$ is a basis of $ V$ with the first $ m$ vectors giving a basis of $ U$. Then, the representing matrix of the transformation $ T$ relative to this basis takes the form

$\displaystyle \begin{pmatrix} A & B \ 0 & C \end{pmatrix}$
where $ A$ is an $ m\times m$ matrix representing the restriction transformation $ T\big\vert _U:U\to U$ relative to the basis $ v_1,\ldots, v_m$.



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"invariant subspace" is owned by rmilson. [ full author list (2) ]
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See Also: linear transformation, invariant


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cyclic subspace (Definition) by CWoo
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Cross-references: transformation, matrix, vectors, basis, well defined, restriction, subspace, vector space, linear transformation
There are 11 references to this entry.

This is version 6 of invariant subspace, born on 2002-02-15, modified 2007-06-09.
Object id is 1962, canonical name is InvariantSubspace.
Accessed 72459 times total.

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AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
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Would like to add a proof by pjohara15 on 2008-01-19 10:17:33
Hello-

I am currently working on a proof of this and would like your permission to add it once I finish it.

PJO
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