I was wondering what the exact relationship between invariant subspaces and a block matrix is?
Is it correct to say: Each diagonal block matrix "creates a vector space decomposition" and vice versa? If this is so, I would be interested in understanding how one gets from the diagonal block matrix to the vector space decomposition, e.g.
$ \begin{pmatrix} 1 & 0&0 \\ 0 & 2 &3 \\ 0 & 4 &5 \\ \end{pmatrix}$. Obvisiously, this one has two blocks(the 1) and this 2×2 term. How does this give me a vector space decomposition?
Best Answer
Working from you example, $$ \begin{pmatrix} 1 & 0&0 \\ 0 & 2 &3 \\ 0 & 4 &5 \\ \end{pmatrix}\begin{pmatrix}x\\0\\0\end{pmatrix}=\begin{pmatrix}x\\0\\0\end{pmatrix}$$ $$ \begin{pmatrix} 1 & 0&0 \\ 0 & 2 &3 \\ 0 & 4 &5 \\ \end{pmatrix}\begin{pmatrix}0\\y\\z\end{pmatrix}=\begin{pmatrix}0\\2y+3z\\4y+5z\end{pmatrix}$$ Notice how the two subspaces are invariant under multiplication by the given matrix. Blocks identify invariant subspaces, which makes for easy direct sum decomposition. There is a lot of good information out there on this topic, here is a good start.