42. Diagonal Matrix

A square matrix whose only non-zero entries lie on the main diagonal:

Often written as .

42.1. Why diagonal matrices are useful

Operations become trivial — they act componentwise:

• Multiplication by vector:
• Powers:
• Inverse (when each ):
• Determinant:
• Trace:
• Eigenvalues: themselves
• Eigenvectors: standard basis vectors

42.2. Special cases

• Identity matrix: all
• Zero matrix (square): all
• Scalar matrix: all for the same scalar (i.e. )

42.3. Where they show up

• Diagonalization: turning a general matrix into a diagonal one via change of basis
• SVD: where is diagonal
• Eigen-decomposition: where is diagonal of eigenvalues