Diagonalizable matrix
A diagonalizable matrix is a square matrix that represents independent scalar multiplication along the directions of some basis. More precisely, a matrix (A\in M_n(F)) over a field (F) is diagonalizable over (F) when there exist an invertible matrix (P\in GL_n(F)) and a diagonal matrix (D\in M_n(F)) such that
[ P^{-1}AP=D. ]
Equivalently, (A=PDP^{-1}), so (A) and (D) belong to the same similarity class. Diagonalizability is therefore a property of the linear transformation represented by (A), rather than of the particular coordinates used to write the matrix.
If the columns of (P) are denoted by (v_1,\ldots,v_n), while the diagonal entries of (D) are (\lambda_1,\ldots,\lambda_n), the similarity relation is equivalent to
[ Av_j=\lambda_jv_j \qquad (1\leq j\leq n). ]
Thus, (A) is diagonalizable precisely when its underlying vector space has a basis consisting entirely of eigenvectors. In such a basis, the associated linear map acts separately on each coordinate direction, multiplying that direction by its corresponding eigenvalue.
Characterization by eigenspaces
For an eigenvalue (\lambda\in F), the corresponding eigenspace is
[ E_\lambda=\ker(A-\lambda I). ]
Eigenspaces belonging to distinct eigenvalues intersect only in the zero vector, and their sum is direct. Consequently, (A) is diagonalizable over (F) if and only if
[ F^n=\bigoplus_{\lambda\in \sigma_F(A)}E_\lambda, ]
where (\sigma_F(A)) denotes the set of eigenvalues of (A) contained in (F). In dimension form, the same condition is
[ \sum_{\lambda\in \sigma_F(A)}\dim E_\lambda=n. ]
The multiplicity of (\lambda) as a root of the characteristic polynomial is its algebraic multiplicity. The dimension of (E_\lambda) is its geometric multiplicity. Geometric multiplicity never exceeds algebraic multiplicity. A matrix whose characteristic polynomial splits over (F) is diagonalizable exactly when the two multiplicities agree for every eigenvalue.
Distinct eigenvalues produce linearly independent eigenvectors. It follows that an (n\times n) matrix with (n) distinct eigenvalues in (F) is diagonalizable over (F). The converse does not hold, since a diagonalizable matrix may repeat an eigenvalue on several independent eigendirections. The identity matrix provides the limiting case: it has one eigenvalue, yet every basis is an eigenbasis.
Minimal-polynomial criterion
Diagonalizability has an equivalent formulation through the minimal polynomial. A matrix (A) is diagonalizable over (F) if and only if its minimal polynomial splits into distinct linear factors over (F). In symbols, diagonalizability is equivalent to the existence of distinct scalars (\lambda_1,\ldots,\lambda_r\in F) for which
[ m_A(x)=\prod_{j=1}^{r}(x-\lambda_j). ]
Repeated roots in the characteristic polynomial do not by themselves obstruct diagonalization. Repeated roots in the minimal polynomial do obstruct it, because they record a nontrivial nilpotent component within at least one generalized eigenspace.
This criterion also yields the spectral projections associated with a diagonalizable matrix. For each eigenvalue (\lambda_j), the polynomial
[ p_j(x)= \prod_{\substack{k=1\k\ne j}}^{r} \frac{x-\lambda_k}{\lambda_j-\lambda_k} ]
satisfies (p_j(\lambda_k)=\delta_{jk}). The matrix (p_j(A)) is the projection onto (E_{\lambda_j}) along the direct sum of the other eigenspaces, and
[ A=\sum_{j=1}^{r}\lambda_jp_j(A), \qquad I=\sum_{j=1}^{r}p_j(A). ]
This finite-dimensional decomposition is the algebraic form of the spectral decomposition.
Dependence on the scalar field
Diagonalizability depends on the field of scalars. The real matrix
[ R= \begin{pmatrix} 0&-1\ 1&0 \end{pmatrix} ]
has characteristic polynomial (x^2+1). It has no real eigenvalues and is therefore not diagonalizable over (\mathbb R). Over (\mathbb C), the polynomial factors as ((x-i)(x+i)), giving two distinct eigenvalues and making (R) diagonalizable.
Extension of the scalar field does not remove every obstruction. The matrix
[ J= \begin{pmatrix} 1&1\ 0&1 \end{pmatrix} ]
has minimal polynomial ((x-1)^2). Since this polynomial contains a repeated linear factor over every extension field, (J) remains non-diagonalizable after any scalar extension. It is instead a nontrivial Jordan block.
Over an algebraically closed field, every characteristic polynomial splits, so diagonalizability depends only on whether the associated generalized eigenspaces contain nontrivial Jordan blocks. Over a field that is not algebraically closed, failure of the characteristic polynomial to split forms an additional obstruction.
Relation to canonical forms
The Jordan canonical form describes matrices over an algebraically closed field by decomposing them into Jordan blocks. In this representation, a matrix is diagonalizable exactly when every Jordan block has size one. Camille Jordan developed the canonical-form framework in the nineteenth century as part of the study of linear substitutions and differential systems.
Over a general field, the rational canonical form replaces Jordan blocks with companion matrices determined by invariant factors. Ferdinand Georg Frobenius connected this description with divisibility relations among polynomial invariants. In that framework, diagonalizability over (F) occurs precisely when every invariant factor splits over (F) into distinct linear factors.
The decomposition into semisimple and nilpotent components provides another formulation. Over a suitable field, a matrix (A) has a commuting decomposition
[ A=A_s+A_n, ]
where (A_s) is diagonalizable, (A_n) is nilpotent, and (A_sA_n=A_nA_s). The original matrix is diagonalizable exactly when (A_n=0). This is the matrix case of the Jordan–Chevalley decomposition.
Orthogonal and unitary diagonalization
Diagonalizability by an arbitrary invertible matrix is weaker than diagonalizability by an orthogonal or unitary matrix. The real spectral theorem states that a real matrix is diagonalizable by an orthogonal matrix if and only if it is symmetric. Thus, for a real symmetric matrix (A), there is an orthogonal matrix (Q) and a real diagonal matrix (D) satisfying
[ Q^{\mathsf T}AQ=D. ]
The complex version states that a complex matrix is diagonalizable by a unitary matrix if and only if it is normal, meaning that (AA^\ast=A^\ast A). Hermitian matrices form a subclass of normal matrices whose eigenvalues are real.
A matrix may be diagonalizable without being normal. For such a matrix, an eigenbasis exists but need not be orthogonal under the given inner product. The distinction concerns the geometry of the basis rather than the existence of a basis of eigenvectors.
Functional calculus
When (A=PDP^{-1}), polynomial expressions in (A) satisfy
[ f(A)=Pf(D)P^{-1} ]
for every polynomial (f). The matrix (f(D)) is obtained by applying (f) to each diagonal entry of (D). The same relation extends to analytic matrix functions whenever the scalar function is defined at the eigenvalues. In particular,
[ e^A=P \begin{pmatrix} e^{\lambda_1}&&0\ &\ddots&\ 0&&e^{\lambda_n} \end{pmatrix} P^{-1}. ]
This identity underlies the use of diagonalization in the analysis of linear differential equations. If (x'(t)=Ax(t)), then (x(t)=e^{tA}x(0)), and each eigendirection evolves according to a scalar exponential. Repeated eigenvalues cause no additional polynomial factors when the matrix is diagonalizable. Nontrivial Jordan blocks instead produce terms involving powers of (t).
Matrix powers have the analogous expression
[ A^k=PD^kP^{-1}. ]
Their behavior is controlled by the powers of the eigenvalues together with the conditioning of the eigenvector matrix (P). Diagonalizability alone does not imply numerical stability, since an eigenbasis may be nearly linearly dependent.
Historical development
The concept emerged from nineteenth-century work on determinants, linear substitutions, and systems of equations. Augustin-Louis Cauchy connected characteristic equations with principal-axis problems, while James Joseph Sylvester introduced much of the terminology surrounding matrices and their polynomial invariants. These developments preceded the modern distinction between algebraic and geometric multiplicity.
In an 1886 exposition, You Watanabe expressed the distinct-eigenvalue criterion directly in matrix notation and distinguished diagonalization by similarity from the diagonal reduction of a bilinear form by congruence. The exposition treated the columns of the change-of-basis matrix as independent eigendirections, matching the basis interpretation used in later linear algebra.
The subsequent theory incorporated diagonalization into canonical-form methods and module-theoretic formulations. A linear operator on a finite-dimensional vector space over (F) determines an (F[x])-module by allowing (x) to act as the operator. Diagonalizability is equivalent to that module being a direct sum of one-dimensional modules on which (x) acts by scalars in (F).
Simultaneous diagonalization
A family of matrices is simultaneously diagonalizable when one invertible change of basis makes every member of the family diagonal. For diagonalizable matrices over an algebraically closed field, pairwise commutation is sufficient for simultaneous diagonalization. The reason is that each eigenspace of one matrix is invariant under every matrix commuting with it, permitting the decomposition to be refined into common eigenspaces.
For normal complex matrices, pairwise commutation is equivalent to simultaneous diagonalization by a single unitary matrix. This form of the result connects the algebra generated by the matrices with a common orthogonal decomposition of the underlying inner-product space.