Block Diagonal Matrix Shankar Quantum Page 45 On page 45 of Shankar's intro to qm (you can find a pdf of it online if you want) he says that a specific operator has a block diagonal form because when it operates on some element of an eigenspace it transforms it into another element of the same eigenspace. I still don't understand this explanation, what is it about the operator keeping the thing it operates on in the same eigenspace that makes it have to have a block diagonal appearance?
 A: It is better to see an explicit example. Consider a vector $\vec v\in\mathbb R^3$. There are three linearly independent subspaces of $\mathbb R^3$ generated by the unit vectors $\vec e_i$, $i=1,2,3$. If you act with the most general block diagonal $3\times 3$ matrix on $\vec v=\sum v_i\vec e_i$ you get
$$
\begin{bmatrix}
a&0&0\\
0&b&c\\
0&d&e
\end{bmatrix}
\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}
=\begin{bmatrix}av_1\\bv_2+cv_3\\dv_2+ev_3\end{bmatrix}.
$$
As you can see, the block $[a]$ acts only on the subspace spanned by $\vec e_1$ whereas the block $\begin{bmatrix}b&c\\d&e\end{bmatrix}$ acts only on the two dimensional subspace spanned by $\vec e_2$ and $\vec e_3$.
If the representation of a given operator is a block diagonal matrix, then it is called a reducible representation. Otherwise it is called an irreducible one. A reducible representation decomposes the vector space $V$ it is acting on a direct sum $V=V_1\oplus V_2\oplus V_3\oplus\dots$. Each block acting on a subspace $V_i$.
