structural:scholar_edition
Applications

Diagonal matrices occur in many areas of linear algebra. Because of the simple description of the matrix operation and eigenvalues/eigenvectors given above, it is typically desirable to represent a given matrix or linear map by a diagonal matrix.

In fact, a given n -by- n matrix A is similar to a diagonal matrix (meaning that there is a matrix X such that X AX is diagonal) if and only if it has n linearly independent eigenvectors. Such matrices are said to be diagonalizable.

Over the field of real or complex numbers, more is true. The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix (if AA = A A then there exists a unitary matrix U such that UAU is diagonal). Furthermore, the singular value decomposition implies that for any matrix A, there exist unitary matrices U and V such that U AV is diagonal with positive entries.