nondiagonal

In linear algebra, nondiagonal describes a square matrix that is not in diagonal form. A diagonal matrix has nonzero entries only on the main diagonal (from the top left to the bottom right) and zeros elsewhere; a nondiagonal matrix has at least one nonzero entry outside the main diagonal.

An example is a 2x2 matrix such as [ [3, 1], [0, 4] ]. This matrix is nondiagonal because the off-diagonal entry 1 is nonzero. In contrast, a diagonal matrix would have zeros in the off-diagonal positions, like [ [3, 0], [0, 4] ].

The distinction matters for computations and structure. Diagonal matrices are especially easy to work with: their eigenvalues are the diagonal entries, powers are obtained by raising the diagonal elements, and many problems reduce to simple scalar operations. Nondiagonal matrices may or may not be diagonalizable. If a matrix is diagonalizable, it can be similar to a diagonal matrix via a change of basis, meaning there exists a basis of eigenvectors. If it is not diagonalizable, it cannot be transformed into a diagonal form through similarity; instead it may be represented in Jordan form, which typically contains nontrivial off-diagonal ones.

In applications, nondiagonal entries often represent coupling, interaction, or connections between components. For example, in physics and engineering, off-diagonal terms in a Hamiltonian or stiffness matrix indicate interactions between states or modes. In graph theory, the off-diagonal entries of an adjacency matrix correspond to edges between vertices.

Overall, nondiagonal is the general descriptor for matrices that are not in diagonal form, with diagonalizability and Jordan form describing deeper structural possibilities.