Diagonalization

A precise guide to diagonalizable matrices, eigenbases, powers, and the role of multiplicity.

Vector-space ideas connect sets, operations, spanning, independence, basis, and dimension. A vector space sits at the centre of a dependency map. Subspace conditions lead to span and independence, which meet at basis and determine dimension; kernel and range connect through a transformation.

Precise definition

A square matrix $A$ is diagonalizable over a field if $A=PDP^{-1}$ for an invertible $P$ and diagonal $D$. Equivalently, the vector space has a basis of eigenvectors of $A$. Columns of $P$ are those eigenvectors, and diagonal entries of $D$ are the corresponding eigenvalues in the same order.

Notation and mathematical language

Algebraic multiplicity is a root's multiplicity in the characteristic polynomial; geometric multiplicity is $\dim\ker(A-\lambda I)$. A matrix is diagonalizable iff the geometric multiplicities sum to $n$, equivalently each eigenvalue has enough independent eigenvectors. Distinct eigenvalues always yield independent eigenvectors.

Conceptual picture

In an eigenbasis, $A$ merely scales each coordinate, so repeated application is easy: $A^k=PD^kP^{-1}$. Diagonalization changes representation, not the underlying linear transformation.

Fully worked example

Interpretation and application

Diagonalization simplifies recurrences, differential systems, Markov chains, and quadratic forms. If eigenvectors are nearly dependent, the formula can be numerically sensitive even when exact diagonalization exists.