Diagonalizable matrix
In linear algebra, a diagonalizable matrix is a square matrix A that is similar to a diagonal matrix, meaning there exists an invertible matrix P such that P^{-1} A P = D, where D is a diagonal matrix whose entries are the eigenvalues of A.[1] This similarity transformation diagonalizes A, effectively representing it in a basis of its eigenvectors, with the diagonal entries of D corresponding to the scaling factors along those basis directions.[2] A matrix A \in \mathbb{R}^{n \times n} (or over the complex numbers) is diagonalizable if and only if it has a full set of n linearly independent eigenvectors, which occurs when the geometric multiplicity equals the algebraic multiplicity for each eigenvalue.[3] Matrices with distinct eigenvalues are always diagonalizable, as each eigenvalue has algebraic multiplicity one and thus a full eigenspace dimension.[4] Not all matrices are diagonalizable; for example, certain Jordan blocks with repeated eigenvalues and deficient eigenspaces are not.[5] Diagonalization is crucial for simplifying matrix computations, such as raising a matrix to a power A^k = P D^k P^{-1}, where D^k is easily computed by raising each diagonal entry to the k-th power.[6] It also facilitates the computation of matrix exponentials e^A = P e^D P^{-1}, which are essential in solving systems of linear differential equations and modeling continuous-time dynamical systems.[7] In applications like Markov chains, principal component analysis, and quantum mechanics, diagonalizable matrices enable efficient spectral decomposition and stability analysis.[8]Fundamentals
Definition
In linear algebra, a square matrix A of size n \times n over a field F is diagonalizable if there exists an invertible matrix P (also of size n \times n) and a diagonal matrix D such that A = P D P^{-1}, where the diagonal entries of D are the eigenvalues of A.[9] This relation expresses A as similar to a diagonal matrix via a change of basis given by the columns of P. The similarity transformation P^{-1} A P = D preserves key spectral properties of A, including its eigenvalues, characteristic polynomial, trace, and determinant, as these are invariant under similarity. For matrices with real entries, the field F is typically extended to the algebraically closed field of complex numbers to guarantee the existence of all eigenvalues, even if they are non-real. The origins of the diagonalization concept trace back to Joseph-Louis Lagrange's 18th-century investigations of quadratic forms, where he employed linear transformations to reduce them to diagonal form, and were formalized within modern linear algebra by David Hilbert's work on spectral theory around 1900.[10]Characterization
A square matrix A over a field F is diagonalizable if and only if there exists a basis of the underlying vector space consisting of n linearly independent eigenvectors of A, where n is the dimension of the space.[11] This condition ensures that A can be represented by a diagonal matrix in some basis, as the eigenvectors form the columns of the invertible matrix P in the similarity transformation A = P D P^{-1}, where D is diagonal.[12] An equivalent characterization involves the multiplicities of the eigenvalues of A. For each eigenvalue \lambda of A, the geometric multiplicity, defined as \dim(\ker(A - \lambda I)), must equal the algebraic multiplicity, which is the multiplicity of \lambda as a root of the characteristic polynomial \det(A - \lambda I).[13] This equality holds across all eigenvalues if and only if the sum of the geometric multiplicities is n, guaranteeing a full basis of eigenvectors.[11] Another criterion uses the minimal polynomial of A, the monic polynomial of least degree that annihilates A. The matrix A is diagonalizable over F if and only if its minimal polynomial factors into distinct linear factors over F, meaning it has no repeated roots.[14] This condition implies that the minimal polynomial splits completely into linear terms without multiplicity greater than one. Over an algebraically closed field such as \mathbb{C}, the characteristic polynomial of any matrix always splits into linear factors by the fundamental theorem of algebra.[15] In this setting, diagonalizability reduces to the minimal polynomial having distinct linear factors or, equivalently, the algebraic and geometric multiplicities matching for each eigenvalue, as the splitting is automatic. For matrices over general fields F, diagonalizability requires both that the characteristic polynomial splits into linear factors over F and that the geometric multiplicity equals the algebraic multiplicity for each root.[15]Diagonalization Techniques
Diagonalization Procedure
To diagonalize a square matrix A \in \mathbb{R}^{n \times n} (or over \mathbb{C}), the procedure involves computing its eigenvalues and eigenvectors to determine if a basis of n linearly independent eigenvectors exists, enabling the decomposition A = PDP^{-1} where D is diagonal and P is invertible.[16]/07%3A_Spectral_Theory/7.02%3A_Diagonalization) The first step is to find the eigenvalues by solving the characteristic equation \det(A - \lambda I) = 0, where I is the identity matrix; the roots \lambda_1, \dots, \lambda_k (with possible multiplicities) are the eigenvalues of A.[16][17] For each distinct eigenvalue \lambda_i, compute the corresponding eigenspace by solving the eigenvector equation (A - \lambda_i I)v = 0 to find a basis for the null space; the dimension of this eigenspace is the geometric multiplicity of \lambda_i.[16]/07%3A_Spectral_Theory/7.02%3A_Diagonalization) The matrix A is diagonalizable if and only if the geometric multiplicity equals the algebraic multiplicity (from the characteristic polynomial) for every eigenvalue, ensuring the eigenspaces collectively span \mathbb{R}^n (or \mathbb{C}^n) with n linearly independent eigenvectors.[16][17] If n linearly independent eigenvectors v_1, \dots, v_n are obtained, form the matrix P with these as columns and the diagonal matrix D = \operatorname{diag}(\lambda_1, \dots, \lambda_n); the diagonalization is then A = PDP^{-1}, which can be verified by direct computation.[16]/07%3A_Spectral_Theory/7.02%3A_Diagonalization) If the geometric multiplicity is less than the algebraic multiplicity for any eigenvalue, the matrix is defective and not diagonalizable over the field; in such cases, the Jordan canonical form provides an alternative block-diagonal representation using generalized eigenvectors, though it requires additional computational steps beyond standard eigendecomposition.[18][19] For large matrices, numerical implementations in software such as MATLAB'seig function or Python's SciPy linalg.eig are essential, as they employ algorithms like the QR method for eigenvalue computation. However, these are sensitive to floating-point errors, particularly for matrices with clustered or nearly degenerate eigenvalues, where small perturbations can lead to inaccurate eigenvectors or failure to detect linear independence.[19][20]