Invariant Subspace¶

Definition¶

Let $\boldsymbol{A}\in\mathcal{L}(V)$ be a linear operator. A subspace $U\subseteq V$ is an invariant subspace under $\boldsymbol{A}$ if $\boldsymbol{A}\boldsymbol{u}\in U$ for every $\boldsymbol{u}\in U$$U$.

Interpretation¶

An invariant subspace is a part of the vector space that the operator never sends outside itself.

The figure uses $\boldsymbol{A}=\begin{pmatrix}2&1\\0&3\end{pmatrix}$$\left(\begin{array}{cc}2& 1\\ 0& 3\end{array}\right)$. The horizontal axis $U=\operatorname{span}(\boldsymbol{e}_1)$ is invariant: a vector $\boldsymbol{u}$ on it is sent to $2\boldsymbol{u}$, which stays on the axis. The vertical axis is not, since $\boldsymbol{A}\boldsymbol{e}_2=(1,3)$ leaves it. The $0$ in the lower left entry of $\boldsymbol{A}$ is what keeps $U$ invariant: it says that $\boldsymbol{A}\boldsymbol{e}_1$ has no $\boldsymbol{e}_2$-component.

A line $\operatorname{span}(\boldsymbol{p})$ with $\boldsymbol{p}\neq\boldsymbol{0}$ is invariant if and only if $\boldsymbol{A}\boldsymbol{p}=\lambda\boldsymbol{p}$ for some $\lambda$, that is, if and only if $\boldsymbol{p}$ is an eigenvector. So the one-dimensional invariant subspaces are exactly the eigenvector lines. Besides the horizontal axis, the figure's matrix has the invariant line through $(1,1)$, which it sends to $(3,3)$. The kernel is invariant because its vectors go to $\boldsymbol{0}$, and the range because every $\boldsymbol{A}\boldsymbol{u}$ lies in it.

If the first $k$ vectors of a basis span an invariant subspace, the first $k$ columns of the matrix in that basis have zeros below row $k$, a block upper triangular shape; $\boldsymbol{A}$ above is the case $k=1$$1$. The span of a Jordan chain is invariant as well, which is why the Jordan normal form splits into separate blocks.

Invariant does not mean unchanged: $\boldsymbol{A}\boldsymbol{u}$ must stay in $U$ but may differ from $\boldsymbol{u}$, as $2\boldsymbol{u}$ does in the figure.

Key Results¶

• $\{\boldsymbol{0}\}$, $V$, $\ker \boldsymbol{A}$, and $\operatorname{range}\boldsymbol{A}$ are basic invariant subspaces.
• If $\boldsymbol{A}\boldsymbol{p}=\lambda \boldsymbol{p}$$\lambda \mathbf{p}$, then $\operatorname{span}(\boldsymbol{p})$ is invariant under $\boldsymbol{A}$.

Examples¶

• For a diagonal matrix, each coordinate axis is invariant.
• The kernel of any linear operator is invariant under that operator.

Prerequisites¶

• Subspace
• Linear Operator

Related Concepts¶

• Eigenvalue and Eigenvector
• Jordan Normal Form
• Generalized Eigenvector

Sources¶

• Math II: Linear Algebra · Week 6

Aliases: stable subspace Symbols: element of, lambda, subset

Last updated: 20 October 2026 03:02