# What is the purpose and dimensionality of expressing a representation of a matrix group as a block diagonal matrix?

The question relates to the following definition:

A decomposition of a representation $$R:G \to GL(n, \mathbb C)$$ is a splitting of $$\mathbb C^n = V_1 \oplus \cdots \oplus V_k$$ where each $$V_i \subseteq \mathbb C^n$$ is a subrepresentation of $$\mathbb C^n$$. Which means that $$R(g) v \in V_i$$ whenever $$v\in V_i.$$ And $$R(g)$$ can be written as a block diagonal matrix:

$$R(g)=\begin{bmatrix}R(g)\vert_{V_i}&&\\ &\ddots& \\ &&R(g)\vert_{V_k}\end{bmatrix}$$ Or, equivalently, that $$R=R\vert_{V_1}\oplus\cdots \oplus R\vert_{V_k}.$$

Leaving aside possible technical misunderstandings, the idea is that the internal direct sum is supposed to change the span of subspaces involved, but not the dimension of the ambient space.

But each block in the matrix above is $$n \times n,$$ which would make the whole matrix $$(kn)\times (kn),$$ and inconsistent with a matrix acting on $$\mathbb C^n$$ vectors.

Is this a correct view of the dimensionality of the block matrix, and if so, what is the purpose of the block matrix expression of this direct sum? In particular, I find it confusing when it comes the the expression $$R(g) v,$$ which seems to say 'the representation applied to the vector $$\vec v$$', as in matrix multiplication, which would call for $$[n\times n][n\times 1]$$ dimensions.

$$R(g)$$ is still a map that sends a vector $$v \in \mathbb{C}^n$$ to a vector $$R(g) v \in \mathbb{C}^n$$.
However, $$R(g)|_{V_i}$$ is not $$n \times n$$; it is $$\dim(V_i) \times \dim(V_i)$$. Since $$\sum_i \dim(V_i) = n$$, there is no discrepancy in the size of the matrix. The block diagonal form of $$R(g)$$ is assuming that we have chosen a basis for $$\mathbb{C}^n$$ by collecting together bases for each $$V_i$$. For example, vectors in $$V_1$$ would have zeros in all entries except the first $$\dim(V_1)$$ entries, so the restriction $$R(g)|_{V_1}$$ of $$R(g)$$ to $$V_1$$ can be represented as a $$\dim(V_1) \times \dim(V_1)$$ matrix.
• So the block matrix is $n \times n$? if $n=5$ and $\mathop dim(V_1)=2,$ the vectors in $V_1$ would be $\begin{bmatrix}a & b &0&0&0\end{bmatrix}^\top,$ and $R(g)|_{V_1}$ can be represented by a $2\times 2$ matrix. Commented Nov 28, 2020 at 23:03