I read the definition of direct sum on wikipedia, and got the idea that a direct sum of two matrices is a block diagonal matrix.
However this does not help me understand this statement in a book. In the book I am reading, the matrix $ \begin{pmatrix} 0&0&0&1 \\ 0&0&1&0 \\ 0&1&0&0 \\ 1&0&0&0 \end{pmatrix} $
"can be regarded as the direct sum of two submatrices":
$ \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix},\begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}$
Where onen lies in the first and fourth rows (columns) and the other in the second and third.
According to the definition it should be
$ \begin{pmatrix} 0&1&0&0 \\ 1&0&0&0 \\ 0&0&0&1 \\ 0&0&1&0 \end{pmatrix} $
This was taken from a problem in Problems and Solutions in Group Theory for Physicists by Zhong-Qi Ma and Xiao-Yan Gu. Here's the problem and the solution in full.
Problem 3. Calculate the eigenvalues and eigenvectors of the matrix $R$ $ R = \begin{pmatrix} 0&0&0&1 \\ 0&0&1&0 \\ 0&1&0&0 \\ 1&0&0&0 \end{pmatrix}. $
Solution. $R$ can be regarded as the direct sum of the two submatrices $\sigma_1$, one lies in the first and fourth rows(columns), the other in the second and third rows(columns). From the result of Problem 2, two eigenvalues of $R$ are $1$, the remaining two are $-1$. The relative eigenvalues are as follows: $ 1: \begin{pmatrix} 1\\ 0 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0\\ 1 \\ 1 \\ 0 \end{pmatrix}, \ \ \ \ \ \ -1: \begin{pmatrix} 1\\ 0 \\ 0 \\ -1 \end{pmatrix}, \begin{pmatrix} 0\\ 1 \\ -1 \\ 0 \end{pmatrix}. $
Problem 2 refers to an earlier problem that calculates the eigenvalues and eigenvectors of the matrix $ \sigma_1= \begin{pmatrix} 0&1 \\ 1&0 \end{pmatrix}. $
[Edit by SN:] Added the full problem text.