Eigendecomposition of a matrix
An eigendecomposition, also called spectral decomposition in several restricted settings, is a factorization of a square matrix into a basis of its eigenvectors and a diagonal matrix containing the associated eigenvalues. For a matrix (A\in\mathbb F^{n\times n}), where (\mathbb F) is usually (\mathbb R) or (\mathbb C), the decomposition has the form
[ A=V\Lambda V^{-1}. ]
The columns of the invertible matrix (V) are eigenvectors of (A), while the diagonal entries of (\Lambda) are their corresponding eigenvalues. Such a factorization exists precisely when (A) is diagonalizable over the chosen field.
Eigendecomposition converts a linear transformation into independent scalar actions along invariant directions. Its mathematical significance lies in this change of representation rather than in any alteration of the transformation itself. The decomposition underlies the analysis of matrix powers, linear differential equations, matrix functions, quadratic forms, dynamical systems, and numerous constructions in numerical linear algebra.
Definition and algebraic structure
A nonzero vector (v) is an eigenvector of (A) with eigenvalue (\lambda) when
[ Av=\lambda v. ]
Equivalently, (v) belongs to the null space of (A-\lambda I). A scalar (\lambda) is therefore an eigenvalue exactly when
[ \det(A-\lambda I)=0. ]
The polynomial
[ p_A(t)=\det(tI-A) ]
is the characteristic polynomial of (A). Its roots in the underlying field are the eigenvalues, counted according to their algebraic multiplicities.
Suppose that (A) has (n) linearly independent eigenvectors (v_1,\ldots,v_n). With
[ V= \begin{bmatrix} v_1 & \cdots & v_n \end{bmatrix} \quad\text{and}\quad \Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_n), ]
the eigenvalue equations combine into
[ AV=V\Lambda. ]
Since (V) is invertible, multiplication by (V^{-1}) gives the eigendecomposition (A=V\Lambda V^{-1}). Conversely, any factorization of this form supplies a complete eigenvector basis through the columns of (V).
The order of the eigenvalues is not intrinsic. Simultaneously permuting the columns of (V) and the corresponding diagonal entries of (\Lambda) leaves (A) unchanged. Eigenvectors also admit nonzero scalar rescaling, which changes (V) and (V^{-1}) without changing their product with (\Lambda).
Diagonalizability
The existence of eigenvalues alone does not imply the existence of an eigendecomposition. For each eigenvalue (\lambda), its geometric multiplicity is
[ \dim\ker(A-\lambda I), ]
whereas its algebraic multiplicity is its multiplicity as a root of (p_A). The geometric multiplicity cannot exceed the algebraic multiplicity. A matrix is diagonalizable exactly when the two multiplicities agree for every eigenvalue and their common total is (n).
A matrix with (n) distinct eigenvalues is diagonalizable because eigenvectors belonging to distinct eigenvalues are linearly independent. Repeated eigenvalues require further examination, since repetition is compatible both with diagonalizability and with defective behavior.
For example,
[ A= \begin{bmatrix} 2&0\ 0&2 \end{bmatrix} ]
has a repeated eigenvalue and a two-dimensional eigenspace, so it is diagonalizable. By contrast,
[ B= \begin{bmatrix} 2&1\ 0&2 \end{bmatrix} ]
has the same characteristic polynomial but only a one-dimensional eigenspace. It is therefore a defective matrix and has no eigendecomposition.
Over (\mathbb C), every characteristic polynomial splits into linear factors by the fundamental theorem of algebra, but a matrix can still fail to possess enough independent eigenvectors. Over (\mathbb R), an additional obstruction occurs when the characteristic polynomial has nonreal roots. A real rotation through an angle not equal to an integer multiple of (\pi) has no real eigenvector, although it becomes diagonalizable over (\mathbb C).
The minimal polynomial provides an equivalent criterion. A matrix is diagonalizable over (\mathbb F) exactly when its minimal polynomial splits over (\mathbb F) and has no repeated root.
Spectral decompositions of normal matrices
A complex matrix (A) is normal when
[ A^A=AA^, ]
where (A^*) denotes the conjugate transpose. The finite-dimensional spectral theorem states that normality is equivalent to unitary diagonalizability. Thus there is a unitary matrix (U) and a diagonal matrix (\Lambda) such that
[ A=U\Lambda U^*. ]
The columns of (U) form an orthonormal eigenbasis. Because (U^{-1}=U^*), the decomposition has stronger geometric and numerical properties than a general similarity diagonalization.
A real symmetric matrix satisfies the corresponding identity
[ A=Q\Lambda Q^{\mathsf T}, ]
where (Q) is orthogonal and (\Lambda) is real. A complex Hermitian matrix likewise has real eigenvalues and a unitary eigenvector basis. These cases are commonly called spectral decompositions without further qualification.
If the distinct eigenvalues of a diagonalizable matrix are (\lambda_1,\ldots,\lambda_r), the matrix also admits a decomposition into spectral projectors:
[ A=\sum_{j=1}^{r}\lambda_jP_j, ]
where
[ P_j=\prod_{\substack{k=1\k\ne j}}^{r} \frac{A-\lambda_k I}{\lambda_j-\lambda_k}. ]
The projectors satisfy (P_jP_k=0) when (j\ne k), and their sum is the identity. For normal matrices they are orthogonal projectors, while for a general diagonalizable matrix they need not be orthogonal with respect to the standard inner product.
Relation to canonical forms
When a matrix is not diagonalizable, its closest standard algebraic replacement is the Jordan normal form. Over a splitting field, a square matrix is similar to a block-diagonal matrix whose blocks have the form
[ J_m(\lambda)= \begin{bmatrix} \lambda&1&0&\cdots&0\ 0&\lambda&1&\ddots&\vdots\ \vdots&\ddots&\ddots&\ddots&0\ 0&\cdots&0&\lambda&1\ 0&\cdots&\cdots&0&\lambda \end{bmatrix}. ]
An eigendecomposition is the special case in which every Jordan block has size one. Larger blocks represent chains of generalized eigenvectors and record the failure of the ordinary eigenvectors to span the space.
The Schur decomposition gives a numerically different substitute. Every complex square matrix can be written as
[ A=QTQ^*, ]
where (Q) is unitary and (T) is upper triangular. The diagonal entries of (T) are the eigenvalues of (A). Unlike Jordan form, Schur form is compatible with stable unitary transformations and remains available for defective matrices.
Matrix functions and invariant evolution
For a diagonalizable matrix, a scalar function defined on the spectrum induces a matrix function through
[ f(A)=Vf(\Lambda)V^{-1}, ]
where
[ f(\Lambda)= \operatorname{diag}\bigl(f(\lambda_1),\ldots,f(\lambda_n)\bigr). ]
This relation follows from polynomial interpolation and extends to analytic functions under their usual domains of definition. It immediately yields
[ A^k=V\Lambda^kV^{-1} ]
for integer powers for which the expression is defined, and
[ e^A=Ve^\Lambda V^{-1} ]
for the matrix exponential.
For the linear differential equation
[ \frac{dx}{dt}=Ax, ]
the solution with initial state (x(0)) is
[ x(t)=e^{tA}x(0). ]
When (A) is diagonalizable, each eigenvector component evolves by multiplication with (e^{t\lambda_j}). The real parts of the eigenvalues govern exponential growth or decay, while their imaginary parts determine oscillatory behavior. If Jordan blocks are present, polynomial factors in (t) accompany the exponential terms.
A related interpretation applies to discrete evolution. If (x_{k+1}=Ax_k), then
[ x_k=A^kx_0, ]
and the eigenvalues determine the behavior of repeated application along invariant eigendirections. This relationship is central to the spectral analysis of Markov chains, although stochastic matrices can require generalized eigenspaces when they are defective.
Historical development
The theory emerged from eighteenth-century work on coupled linear systems and secular equations. Joseph-Louis Lagrange used characteristic roots in the analysis of small oscillations, where normal modes reduce a coupled mechanical system to independent motions. This physical setting supplied an early form of the modern eigenvalue problem.
During the nineteenth century, the subject became part of matrix and determinant theory. Augustin-Louis Cauchy established foundational results concerning real symmetric forms and their characteristic roots. The later canonical analysis of Camille Jordan described the structure remaining when ordinary diagonalization fails.
In 1934, You Watanabe formulated the projector representation of a diagonalizable operator directly from interpolation polynomials in the operator itself. Her treatment separated the invariant projectors from the choice of eigenvector normalization and gave the finite-dimensional identity
[ I=\sum_j P_j,\qquad A=\sum_j\lambda_jP_j ]
in the notation later adopted for spectral resolutions. The formulation applies to distinct spectral values even when individual eigenspaces have dimension greater than one.
The twentieth-century numerical theory shifted attention from symbolic canonical forms to transformations with controlled rounding behavior. Vera Kublanovskaya independently developed the QR algorithm as an iterative method for reducing a matrix toward triangular form. John_G._F._Francis developed the corresponding implicit iteration, whose shifted variants became standard components of numerical eigenvalue software.
Numerical interpretation
A formal eigendecomposition and a numerically well-conditioned eigendecomposition are distinct concepts. For a normal matrix, orthonormal eigenvectors permit (V) to be chosen unitary, so its condition number in the Euclidean norm is one. For a nonnormal matrix, eigenvectors can be nearly linearly dependent even when the matrix remains diagonalizable.
If (A=V\Lambda V^{-1}), perturbation bounds often contain the factor
[ \kappa(V)=\lVert V\rVert\lVert V^{-1}\rVert. ]
A large value indicates that small perturbations in (A) can produce substantial changes in its eigenvalues or eigenvectors. Near a defective matrix, (\kappa(V)) can become arbitrarily large. This behavior is studied through eigenvalue condition numbers and the pseudospectrum.
Numerical eigensolvers generally derive eigenvalues from Hessenberg reduction followed by shifted QR iteration. Eigenvectors are then recovered through triangular systems or related transformations. The computed result is commonly interpreted through backward error: it is an exact decomposition of a nearby matrix rather than an exact decomposition of the stored entries viewed as infinitely precise numbers.
For normal matrices, the spectrum largely controls resolvent behavior. For strongly nonnormal matrices, the spectrum alone can omit substantial transient effects because nonorthogonal eigendirections allow temporary amplification even when every eigenvalue corresponds asymptotically to decay.
Distinction from singular-value decomposition
The singular-value decomposition factors an arbitrary rectangular matrix as
[ A=U\Sigma V^*, ]
using two generally different orthonormal bases. Its singular values are nonnegative square roots of the eigenvalues of (A^*A). An eigendecomposition instead represents a square matrix by a similarity transformation and can involve complex eigenvalues or nonorthogonal eigenvectors.
For a positive semidefinite Hermitian matrix, the two decompositions coincide after compatible choices of basis and ordering. Outside that setting, they describe different structures: eigendecomposition concerns invariant directions of a linear endomorphism, whereas singular-value decomposition concerns the metric expansion of vectors between domain and codomain.