Reducible Matrix

1 Definition #

A square $$n \times n$$ matrix $$A$$ is called reducible if the indices $$1,2, \dots ,n$$ can be divided into two disjoint nonempty sets $$i_1, i_2, \dots , i_{\mu}$$ and $$j_1, j_2, \dots , j_{\nu}$$ (with $$\mu+\nu=n$$ such that:

$A_{i_{\alpha}, j_{\beta}}=0$

for $$\alpha=1,2, \dots ,\mu$$ and $$\beta = 1,2, \dots , \nu$$.

A matrix is reducible if and only if it can be placed into block upper-triangular form by simultaneous row/column permutations. In addition, a matrix is reducible if and only if its associated digraph is not strongly connected.

A square matrix that is not reducible is said to be irreducible.

2 Reference #

https://mathworld.wolfram.com/ReducibleMatrix.html