## 矩阵-不可约表示

irreducible representations: the basic building block for symmetry of the point group.

block diagonalization.

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.

【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/

What is an example of an irreducible matrix?

Markov Chains-整个系列