Linear Algebra 65 | Diagonalizable Matrices [dark version]

Diagonalizable matrices are exactly the square matrices that admit a full set of eigenvectors—enough to rebuild every vector in the space—so the matrix action can be rewritten as a simple scaling along independent directions. That matters because it turns complicated linear transformations into a diagonal form, making powers of the matrix, solving linear systems, and understanding dynamics far easier.

The discussion starts from the idea of changing coordinates. If a matrix A has eigenvectors X1 through Xn that form a basis of C^n, then any vector U can be expressed as a linear combination of those eigenvectors. In that eigenvector coordinate system, A acts by scaling each eigenvector by its corresponding eigenvalue. Writing the eigenvectors as columns of a matrix X, the transformation becomes A = X D X^{-1}, where D is diagonal and its diagonal entries are the eigenvalues (counted with multiplicity). The key question becomes whether such an eigenvector basis exists at all.

That existence question is reframed into a basis test: the eigenvectors span C^n if and only if they are linearly independent and therefore form a basis. Equivalently, the matrix X built from those eigenvectors must be invertible. This leads to the definition: a square matrix is diagonalizable if it has n linearly independent eigenvectors, i.e., if there exists an invertible X and a diagonal D such that A = X D X^{-1}.

Examples clarify when diagonalization works and when it fails. A diagonal matrix is automatically diagonalizable because the standard basis vectors are eigenvectors. A non-diagonal triangular matrix can still be diagonalizable: in the 2×2 case, even if one standard basis vector is not an eigenvector, the matrix may still have two independent eigenvectors, allowing a diagonal form. But diagonalization can break down when eigenvectors are too scarce. For a 2×2 triangular matrix with only one eigenvalue and only a one-dimensional eigenspace, there aren’t enough independent eigenvectors to span C^2, so the matrix is not diagonalizable.

The transcript then gives practical criteria. For each eigenvalue, algebraic multiplicity counts how many times it appears as a root of the characteristic polynomial, while geometric multiplicity counts the dimension of the eigenspace. A matrix is diagonalizable precisely when, for every eigenvalue, geometric multiplicity equals algebraic multiplicity (and the geometric multiplicities sum to n). A major consequence follows: every normal matrix—such as a selfadjoint (Hermitian) matrix—is diagonalizable, and even admits an orthonormal eigenbasis.

Finally, a convenient shortcut is offered: if a matrix has n distinct eigenvalues in an n-dimensional space, then it is diagonalizable. In that situation each eigenvalue has algebraic multiplicity one, and the eigenspaces automatically provide enough independent eigenvectors without explicitly computing them.