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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.

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    I don't understand the question. This matrix is not block diagonal.2011-10-04
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    @Qia So it is not a direct sum? In which case my source is wrong.2011-10-04
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    Did you take this example from somewhere? If so, it might help to quote the relevant text directly, or give a reference/link.2011-10-04
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    @Srivastan For the [Source](http://www.amazon.com/Problems-Solutions-Group-Theory-F/dp/9812388338/ref=sr_1_2?ie=UTF8&qid=1317754556&sr=8-2) click the given link, click the amazon "Look Inside" feature, click on "first pages", check problem 3.2011-10-04
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    @kuch I took the liberty of adding the problem and the solution from the book in full, so that others get the context immediately. Hope it's ok. :)2011-10-04

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