irreducible representations: the basic building block for symmetry of the point group.
block diagonalization.
【可约矩阵】(reducible matrix): An \(n \times n\) matrix \(A\) is said to be a reducible matrix if and only if for some permutation matrix \(P\), the matrix \(P^T A P\) is block upper triangular.
The following conditions on an \(n \times n\) matrix \(A\) are equivalent:
- \(A\) is an irreducible matrix.
- The digraph associated to \(A\) is strongly connected.
- For each \(i\) and \(j\), there exists some \(k\) such that \(\left(A^k\right)_{i j}>0\).
- For any partition \(J \sqcup K\) of the index set \(\{1,2, \ldots, n\}\), there exist \(j \in J\) and \(k \in K\) such that \(a_{j k} \neq 0\).
For certain applications, irreducible matrices are more useful than reducible matrices. In particular, the Perron-Frobenius theorem gives more information about the spectra of irreducible matrices than of reducible matrices.
【不可约矩阵】(irreducible matrix): A matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size). (Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible if and only if such directed graph is strongly connected.) A detailed definition is given here.
矩阵不可约等价于强连通。
【强连通】(strongly connected)：在有向图的数学理论中，如果一个图的每一个顶点都可从该图其他任意一点到达，则称该图是强连通的。在任意有向图中能够实现强连通的部分我们称其为强连通分量。
https://andrewei1316.github.io/2016/04/06/Connectivity-of-Graphs/
参考资料：
adjacency matrix
许少鸿
What is an example of an irreducible matrix?