Invariance

A possible relation between subspaces $ℳ$ of a vector space and linear transformations $A$ on that space is invariance. We say that $ℳ$ is invariant under $A$ , if $x$ in $ℳ$ implies that $Ax$ is in $ℳ$ . (Observe that the implication relation is required in one direction only; we do not assume that every $y$ in $ℳ$ can be written in the form $y=Ax$ with $x$ in $ℳ$ ; we do not even assume that $Ax$ in $ℳ$ implies $x$ in $ℳ$ . Presently we shall see examples in which the conditions we did not assume definitely fail to hold.) We know that a subspace of a vector space is itself a vector space; if we know that $ℳ$ is invariant under $A$ , we may ignore the fact that $A$ is defined outside $ℳ$ and we may consider $A$ as a linear transformation defined on the vector space $ℳ$ . Invariance is often considered for sets of linear transformations, as well as for a single one; $ℳ$ is invariant under a set if it is invariant under each member of the set.

What can be said about the matrix of a linear transformation $A$ on an $n$ -dimensional vector space $𝒱$ if we know that some $ℳ$ is invariant under $A$ ? In other words: is there a clever way of selecting a basis $𝒳=\{x_1,\dots ,x_n\}$ in $𝒱$ so that $[A]=[A;𝒳]$ will have some particularly simple form? The answer is in Section: Dimension of a subspace , Theorem 2; we may choose $𝒳$ so that $x_1,\dots ,x_m$ are in $ℳ$ and $x_{m+1},\dots ,x_n$ are not. Let us express $A{x}_j$ in terms of $x_1,\dots ,x_n$ . For $m+1\le j\le n$ , there is not much we can say: $A{x}_j=\sum _i{\alpha }_{ij}{x}_i$ . For $1\le j\le m$ , however, $x_j$ is in $ℳ$ , and therefore (since $ℳ$ is invariant under $A$ ) $A{x}_j$ is in $ℳ$ . Consequently, in this case $A{x}_j$ is a linear combination of $x_1,\dots ,x_m$ ; the ${\alpha }_{ij}$ with $m+1\le i\le n$ are zero. Hence the matrix $[A]$ of $A$ , in this coordinate system, will have the form where $[A_1]$ is the ( $m$ -rowed) matrix of $A$ considered as a linear transformation on the space $ℳ$ (with respect to the coordinate system $\{x_1,\dots ,x_m\}$ ), $[A_2]$ and $[B_0]$ are some arrays of scalars (in size $(n-m)$ by $(n-m)$ and $m$ by $(n-m)$ respectively), and $[0]$ denotes the rectangular ( $(n-m)$ by $m$ ) array consisting of zeros only. (It is important to observe the unpleasant fact that $[B_0]$ need not be zero.)