The process of finding a basis in which a linear operator or matrix is represented by a diagonal matrix, equivalently finding a full set of eigenvectors so that the operator acts as scalar multiplication on each basis vector.

Principle

A linear operator is diagonalizable over a field if and only if the underlying vector space admits a basis of eigenvectors; algebraically this often corresponds to the minimal (or characteristic) polynomial splitting into linear factors with geometric multiplicities matching algebraic multiplicities, with field-dependence critical.

Demonstration

The matrix [[2,0],[0,3]] is already diagonal and thus diagonalizable with eigenvalues 2 and 3. The Jordan block [[1,1],[0,1]] is not diagonalizable over any field because it lacks two independent eigenvectors. A symmetric real matrix is diagonalizable by an orthogonal change of basis.

Misapplication

Assuming a matrix is diagonalizable over R when its eigenvalues are complex, or conflating diagonalizability with being symmetric: symmetry implies diagonalizability (over R) but diagonalizability does not imply symmetry; incorrectly treating defective eigenvalue multiplicities as harmless.

Consequence

When diagonalizable, many tasks simplify: computing powers and exponentials of the operator reduces to scalar operations on eigenvalues; systems decouple into independent scalar equations; spectral decompositions clarify dynamics and stability properties.

Reversal

Failure of diagonalization leads to Jordan canonical form or other canonical triangular forms; triangularization is a weaker normal form that always exists over algebraically closed fields but retains coupling between generalized eigenspaces.

Boundary

Diagonalizability depends on the base field and on the existence of a complete eigenbasis; multiplicity defects, inseparable polynomials in positive characteristic, and nonalgebraically closed fields can prevent diagonalization. Even when diagonalizable, the diagonal form is defined up to permutation of diagonal entries and scaling of eigenvectors.

Semantic Tension

Tension between diagonalization as algebraic diagonal form and spectral decomposition with orthogonality or unitary structure: normal or symmetric operators admit orthonormal eigenbases, strengthening diagonalization; general diagonalization may require nonorthogonal bases.

Synthesis

Diagonalization identifies a coordinate system of eigenvectors in which a linear operator acts by simple scalar multiplications, drastically simplifying analysis when possible; when impossible, one resorts to Jordan or triangular forms that reveal the residual coupling among generalized eigenvectors.