What is invariant subspace?

An invariant subspace is a subspace of a vector space that is preserved under a linear transformation. More precisely, if V is a vector space and T: V → V is a linear transformation, then a subspace W ⊆ V is called T-invariant if T(w) ∈ W for all w ∈ W.

This means that any vector in the subspace, when acted upon by the linear transformation, remains in the subspace. Invariant subspaces are important in linear algebra because they help to understand the structure of linear operators.

One important theorem about invariant subspaces is the Cayley-Hamilton theorem, which states that every linear operator satisfies its own characteristic equation. This has important consequences for invariant subspaces because it implies that any polynomial in the linear operator can be expressed as a linear combination of lower degree polynomials in the same operator.

Invariant subspaces also play a key role in representation theory, which is the study of symmetries and transformations of objects in mathematics. In this context, invariant subspaces correspond to subspaces that are unchanged under a given group action.

Overall, invariant subspaces are a fundamental concept in linear algebra and have applications across many areas of mathematics and physics.