10 - Block Matrix Decomposition of a Linear Map

Block Matrix Decomposition

Let $V,W$ be finite dimensional vector spaces with $A\subseteq V$ and $B\subseteq W$
Let $\mathcal{B}=\{e_1,\cdots ,e_n\}$ be a basis for $V$ and $\mathcal{E}=\{e_1,\cdots ,e_k\}$ a basis for $A$ (with $k\le n$)
Let ${\mathcal{B}}^{\prime }=\{e_1^{\prime },\cdots ,e_m^{\prime }\}$ be a basis for $W$ and ${\mathcal{E}}^{\prime }=\{e_1^{\prime },\cdots ,e_l^{\prime }\}$ a basis for $B$

So we have induced bases for $V/A$ and $W/B$ given by

$$\begin{array}{rl}\overline{\mathcal{B}}& =e_{k+1}+A,\cdots ,e_n+A\\ \overline{{\mathcal{B}}^{\prime }}& =e_{l+1}^{\prime }+B,\cdots ,e_m^{\prime }+B\end{array}$$

Let $T:V\to W$ be a linear map such that $T(A)\subseteq B$ then $T$ induces a map $\overline{T}$ on quotients and restricts to a linear map

$$T{\mid }_A:A\to B\text{ with }T{\mid }_A(v)=T(v)\text{ for }v\in A$$

Using notation

$${}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}=(a_{ij}{)}_{1\le i\le m,1\le j\le n}$$

so

$$T(e_j)=a_{1j}{e}_1^{\prime }+\cdots +a_{mj}{e}_m^{\prime }$$

Then there is block matrix decomposition

$${}_{{\mathcal{B}}^{\prime }}[T{]}_{\mathcal{B}}=\left(\begin{array}{cc}{}_{{\mathcal{E}}^{\prime }}[T{\mid }_A{]}_{\mathcal{E}}& \ast \\ 0& {}_{\overline{{\mathcal{B}}^{\prime }}}[\overline{T}{]}_{\overline{\mathcal{B}}}\end{array}\right)$$

where ${}_{\overline{{\mathcal{B}}^{\prime }}}[\overline{T}{]}_{\overline{\mathcal{B}}}=(a_{ioj}{)}_{l+1\le i\le m,k+1\le j\le n}$

Proof

For $j\le k$ then $T(e_j)\in B$ so $a_{ij}=0$ for $i>l$
$a_{ij}$ is equal to $(i,j)-$th entry of ${}_{{\mathcal{E}}^{\prime }}[T{\mid }_A{]}_{\mathcal{E}}$ for $i\le l$

For the bottom right corner of the matrix

$$\begin{array}{rl}\overline{T}(e_j+A)& =T(e_j)+B\\ & =a_{1j}{e}_1^{\prime }+\cdots +a_{mj}{e}_m^{\prime }+B\\ & =a_{l+1,j}(e_{l+1}^{\prime }+B)+\cdots +a_{mj}(e_m^{\prime }+B)\end{array}$$