What Is an Invariant Subspace?

October 24, 2023

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A subspace $S$ of $\mathbb{C}^n$ is an invariant subspace for $A\in\mathbb{C}^{n \times n}$ if $AS \subseteq S$ , that’s, if $x\in S$ implies $Ax \in S$ .

Listed below are some examples of invariant subspaces.

$\{0\}$ and $\mathbb{C}^n$ are trivially invariant subspaces of any $A$ .
The null house $\mathrm{null}(A) = \{ x: Ax = 0\}$ is an invariant subspace of $A$ as a result of $x\in\mathrm{null}(A)$ implies $Ax = 0\in\mathrm{null}(A)$ .
If $x$ is an eigenvector of $A$ then $\mathrm{span}(x) = \{\, \alpha x: \alpha\in\mathbb{C} \,\}$ is a $1$ -dimensional invariant subspace, since $A\alpha x = \lambda \alpha x \in S$ , the place $\lambda$ is the eigenvalue comparable to $x$ .
The matrix

$\notag \begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$

has a one-dimensional invariant subspace $\mathrm{span}(e_1)$ and a two-dimensional invariant subspace $\mathrm{span}(e_1, e_2)$ , the place $e_i$ denotes the $i$ th column of the identification matrix.

Let $x_1,x_2, \dots,x_p\in\mathbb{C}^n$ be linearly unbiased vectors. Then $S = \mathrm{span}(x_1,x_2, \dots,x_p)$ is an invariant subspace of $A$ if and provided that $Ax_i\in S$ for $i=1\colon p$ . Writing $X = [x_1,x_2,\dots,x_p]\in\mathbb{C}^{n \times p}$ , this situation might be expressed as

$\notag AX = XB, \qquad(1)$

for some $B\in\mathbb{C}^{p \times p}$ .

If $p = n$ in (1) then $AX = XB$ with $X$ sq. and nonsingular, so $X^{-1}AX = B$ , that’s, $A$ and $B$ are comparable.

Eigenvalue Relations

We denote by $\Lambda(A)$ the spectrum (set of eigenvalues) of $A$ and by $A^+$ the pseudoinverse of $A$ .

Theorem.

Let $A\in\mathbb{C}^{n\times n}$ and suppose that (1) holds for some full-rank $X\in\mathbb{C}^{n\times p}$ and $B\in\mathbb{C}^{p\times p}$ . Then $\Lambda(B)\subseteq\Lambda(A)$ . Moreover, if $(\lambda,x)$ is an eigenpair of $A$ with $x\in\mathrm{range}(X)$ then $(\lambda,X^+x)$ is an eigenpair of $B$ .

Proof. If $(\lambda,z)$ is an eigenpair of $B$ then $AXz = XBz = \lambda Xz$ , and $X z\ne 0$ because the columns of $X$ are unbiased, so $(\lambda,Xz)$ is an eigenpair of $A$ .

If $(\lambda,x)$ is an eigenpair of $A$ with $x\in\mathrm{range}(X)$ then $x = Xz$ for some $z\ne0$ , and $z = X^+x$ , since $X$ being full rank implies that $X^+X = I$ . Therefore

$\notag \lambda x = Ax = AXz = XBz.$

Multiplying on the left by $X^+$ offers $\lambda z = Bz$ , so $(\lambda,z)$ is an eigenpair of $B$ .

Block Triangularization

Assuming that $X$ in (1) has full rank $p$ we will select $Y\in\mathbb{C}^{p \times (n-p)}$ in order that $W = [X,\,Y]$ is nonsingular. Let $W^{-1} = \left[\begin{smallmatrix} G \\ H \end{smallmatrix}\right]$ and be aware that $W^{-1}W = I$ implies $GX = I$ and $HX = 0$ . Now we have

$\notag W^{-1}AW = \begin{bmatrix} G \\H \end{bmatrix} [AX,\, AY] = \begin{bmatrix} G \\H \end{bmatrix} [XB,\, AY] = \begin{bmatrix} B & GAY\\ 0 & HAY \end{bmatrix}, \qquad (2)$

which is block higher triangular. This development is used within the proof of the Schur decomposition with $p=1$ , $x$ an eigenvector of unit $2$ -norm, and $W$ chosen to be unitary.

The Schur Decomposition

Suppose $A\in\mathbb{C}^{n \times n}$ has the Schur decomposition $Q^*AQ = R$ , the place $Q$ is unitary and $R$ is higher triangular. Then $AQ = QR$ and writing $Q = [Q_1,\,Q_2]$ , the place $Q_1$ is $n\times p$ , and

$\notag R = \begin{bmatrix} R_{11} & R_{12} \\ 0 & R_{22} \\ \end{bmatrix},$

the place $R_{11}$ is $p\times p$ , we now have $A Q_1 = Q_1 R_{11}$ . Therefore $Q_1$ is an invariant subspace of $A$ comparable to the eigenvalues of $A$ that seem on the diagonal of $R_{11}$ . Since $p$ can take any worth from $1$ to $n$ , the Schur decomposition supplies a nested sequence of invariant subspaces of $A$ .

Notes and References

Many books on numerical linear algebra or matrix evaluation include materials on invariant subspaces, for instance

David S. Watkins. Fundamentals of Matrix Computations Third version, Wiley, New York, USA, 2010.

The final word reference is maybe the e-book by Gohberg, Lancaster, and Rodman, which has an accessible introduction however is usually on the graduate textbook or analysis monograph stage.

Israel Gohberg, Peter Lancaster, and Leiba Rodman, Invariant Subspaces of Matrices with Functions, Society for Industrial and Utilized Arithmetic, Philadelphia, PA, USA, 2006 (unabridged republication of e-book first printed by Wiley in 1986).

Associated Weblog Posts

This text is a part of the “What Is” collection, obtainable from https://nhigham.com/index-of-what-is-articles/ and in PDF kind from the GitHub repository https://github.com/higham/what-is.

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What Is an Invariant Subspace?

Eigenvalue Relations

Block Triangularization

The Schur Decomposition

Notes and References

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