TITLE: How does one show a matrix is irreducible and reducible?
QUESTION [17 upvotes]: How does one show a matrix is irreducible and reducible? An example would also be great.
I know that a matrix is reducible if and only if it can be placed into block upper-triangular form. How do you find block upper-triangular form?

REPLY [4 votes]: Let $A=[a_{i,j}]\in M_n(\mathbb{R})$ and $|A|=[|a_{i,j}|]$. $A$ is irreducible IFF $|A|$ is too. Then we may assume that the $a_{i,j}$ are $\geq 0$. We have a look at the complexity of the problem: "decide whether $A$ is irreducible or not".
Of course, we do not look for a permutation of the basis vectors that triangularizes $A$ (the complexity is $O(n!)$).
We can use the test (cf. Renko Usami) $(I+A)^{n-1}>0$. Yet, the complexity is $O(n^3\log(n))$.
The best is to use the "strong component algorithm" (cf user1551). Its complexity is $O(n)$ (that is extraordinary fast, even for a $10^6\times 10^6$ matrix).