If $A \in M_n(\mathbb{C})$, then $A$ is called *reducible* if there is a permuation matrix $P$ such that
$$
P^\top A P =
\begin{bmatrix}
A_{11} & A_{12} \\
0 & A_{22}
\end{bmatrix}, $$
in which $A_{11}$ and $A_{22}$ are square matrices of order at least one. If $A$ is not reducible, then $A$ is called *irreducible*. Notice that, with this definition, every one-by-one matrix is irreducible. A matrix $A$ is irreducible if and only if its *directed graph* or *digraph* is *strongly connected*.

It is known (see, e.g., Brualdi and Ryser [Theorem 3.2.4; MR1130611]) that if $A \in M_n(\mathbb{C})$, then $A$ is irreducible or there is a permutation matrix $P$ such that
\begin{equation}
P^\top A P =
\begin{bmatrix}
A_{11} & \cdots & A_{1k} \\
& \ddots & \vdots \\
& & A_{kk}
\end{bmatrix}, \tag{1} \label{fnf}
\end{equation}
in which the matrices $A_{11},\dots, A_{kk}$ are square, irreducible matrices. The matrix in \eqref{fnf} is called a *Frobenius* or *irreducible normal form (of $A$)* (FNF) and it is not unique. However, the blocks $A_{11},\dots, A_{kk}$ are unique up to permutation similarity.

A matrix $A$ is nonnegative ($A \ge 0$) if $a_{ij} \ge 0$, $1 \le i,j \le n$. A nonnegative matrix is *stochastic* if $Ae = e$ and *doubly stochastic* if, in addition, $A^\top e = e$ (i.e., $A$ has row and column sums equal to one).

In a 1965 paper, Perfect and Mirsky [MR0175917] state, without proof, that if $A$ is a doubly stochastic matrix, then every FNF of $A$ is of the form $$ \begin{bmatrix} A_{11} & & \\ & \ddots & \\ & & A_{kk} \end{bmatrix}, $$ i.e., every doubly stochastic matrix is either irreducible or is permutationally similar to a direct sum of irreducible, doubly stochastic matrices. After giving the result Perfect and Mirsky state:

"This result is almost certainly well-known. As, furthermore, it follows very easily from the definitions, we omit the details of the proof." – p. 38.

This result is easy to establish for matrices of order less than or equal to four, but does not seem obvious or seem to follow simply from the definitions above.

**Question 1:** Is this result obvious or is there a simple proof of this fact?

**Question 2:** Does anybody know of a reference?