Invariant

The property for a mathematical property to remain unchanged after operations. For example, the area of a triangle remains the same regardless of rotations or flips.

Definition

1. With $T\in \mathcal{L}(V)$
2. With $W\subset V$ be a Subspace of $V$
3. $T$ is invariant if $T(v)\in W,\forall v\in W$

Intuition

Anything that goes through this linear map will remain fenced in the same domain.

Theorems

• Invariant Subspace Characterization of Triangularability
• Commuting Operators Provide Invariant Subspace
• Invariant Subspaces Give Block Diagonal Matrix Representations