Chapter 7 - Eigenvalues and Eigenvectors - 7.3 Symmetric Matrices and Orthogonal Diagonalization - 7.3 Exercises - Page 371: 42

orthogonally diagonalizable.

A matrix is symmetric if it is equal to its transpose. Also, a matrix is orthogonally diagonalizable if and only if it is symmetric. Here the transpose of the matrix is $\begin{bmatrix} 0&1&0&-1 \\ 1&0&-1& 0\\ 0&-1&0&-1\\ -1&0&-1&0\\ \end{bmatrix} $, which is the same as the original matrix; thus it is symmetric and it is orthogonally diagonalizable.