1 Definition and basic properties
1.1 Formal definition
Let \(A\) be a complex square matrix and let \(A^*\) denote its conjugate transpose (adjoint). The matrix \(A\) is called non-normal if it does not commute with its adjoint, i.e. \[ AA^* \neq A^*A. \] If instead \(AA^* = A^*A\), then \(A\) is normal.
1.2 Equivalent characterizations
1.2.1 Commutator with the adjoint
A convenient formulation uses the commutator \[ [A,A^*] := AA^* - A^*A. \] Then \(A\) is non-normal exactly when \([A,A^*]\neq 0\). This highlights that non-normality is not merely about eigenvalues, but about how \(A\) interacts with the inner product structure through \(A^*\).
1.2.2 Failure of orthogonal diagonalizability
A central characterization is that a matrix \(A\) is normal if and only if it is unitarily diagonalizable. Consequently, non-normal matrices are those that cannot, in general, be diagonalized by a unitary change of basis; their eigenvectors need not form an orthonormal set, and diagonalizability by similarity may still fail or lead to strongly non-orthogonal eigenstructures.
1.3 Examples of non-normal matrices
Common sources of non-normality include matrices that resemble triangular or shift-like operators and those derived from discretizations of differential equations.
- Jordan block (non-normal): The strictly upper triangular Jordan block \(J=\begin{pmatrix}0&1\\0&0\end{pmatrix}\) satisfies \(JJ^*\neq J^*J\), so it is non-normal.
- Upper triangular with distinct diagonal entries: Many upper triangular matrices with more than one nonzero superdiagonal entry are non-normal even though their eigenvalues are easy to read from the diagonal.
- Non-symmetric discretizations: Finite-difference or finite-volume discretizations of advection-dominated problems often produce matrices that are non-normal, reflecting directional transport and boundary effects.
2 Relationship to normal matrices
2.1 Classes of normal matrices
2.1.1 Hermitian matrices
A Hermitian matrix satisfies \(A^*=A\). Hermitian matrices are normal, so they admit a unitary diagonalization and have orthonormal eigenvectors.
2.1.2 Unitary matrices
A unitary matrix satisfies \(A^*A=I\) (equivalently \(AA^*=I\)). Unitary matrices are normal and preserve the Euclidean norm.
2.1.3 Skew-Hermitian matrices
A skew-Hermitian matrix satisfies \(A^*=-A\). It is normal and can be handled similarly to Hermitian matrices after multiplication by \(i\).
2.2 Distinguishing features of non-normal matrices
Normal matrices have tightly structured behavior: their eigenvectors can be chosen orthonormal (after a unitary similarity), and many quantities depend on eigenvalues in a more transparent way. Non-normal matrices lack this orthogonality, so they can show phenomena where transient behavior, sensitivity to perturbations, or growth in norms is dominated by the geometry of eigenvectors and the pseudospectral landscape rather than by the eigenvalues alone.
2.3 Matrix decompositions
2.3.1 Schur decomposition
Every complex square matrix \(A\) admits a Schur decomposition \[ A = U T U^*, \] where \(U\) is unitary and \(T\) is upper triangular. For normal matrices, \(T\) can be taken diagonal; for non-normal matrices, the upper triangular part typically contains significant off-diagonal structure that reflects non-normal coupling between invariant subspaces.
2.3.2 Jordan canonical form
Over \(\mathbb{C}\), matrices can be put into Jordan form via similarity: \[ A = S J S^{-1}, \] where \(J\) is block diagonal with Jordan blocks. Non-normal matrices commonly appear with Jordan blocks larger than \(1\times 1\), indicating non-diagonalizability or, even when diagonalizable, often indicating that the similarity transform \(S\) is ill-conditioned and eigenvectors are far from orthogonal.
3 Spectral properties
3.1 Eigenvalues and eigenvectors
Eigenvalues of any matrix are roots of the characteristic polynomial, and they still determine important aspects of the dynamics of \(A\). However, for non-normal matrices the relationship between eigenvalues and eigenvectors is more delicate:
- eigenvectors may be incomplete if \(A\) is not diagonalizable;
- even when diagonalizable, eigenvectors can be nearly linearly dependent;
- left and right eigenvectors generally fail to be orthogonal or well-conditioned.
Thus eigenvalues alone may not summarize the behavior of \(A\), especially for functions of \(A\) such as exponentials or resolvents.
3.2 Spectral sensitivity
3.2.1 Perturbation of eigenvalues
Small perturbations \(A\mapsto A+\Delta A\) can produce large eigenvalue movements when the eigenstructure is ill-conditioned. The sensitivity is strongly affected by the angle between right and left eigenvectors. For non-normal matrices, this sensitivity can be dramatic even under perturbations of modest norm.
3.2.2 Eigenvector non-orthogonality
For non-normal \(A\), eigenvectors associated with different eigenvalues need not be orthogonal and may be highly non-orthogonal. A typical consequence is that projections onto eigenvector directions amplify errors, making computed eigenvalues and eigenvectors sensitive to numerical inaccuracies and rounding.
3.3 Pseudospectra
The pseudospectrum of \(A\) broadens the notion of spectrum by examining where the resolvent \((zI-A)^{-1}\) becomes large. For non-normal matrices, pseudospectra can extend far beyond the eigenvalues, indicating that the matrix behaves as if it had “effective eigenvalues” in regions where the resolvent norm is high. This often explains transient growth and non-asymptotic sensitivity.
4 Numerical and applied implications
4.1 Stability analysis
4.1.1 Transient growth
| In dynamical systems governed by \(\dot{x}=Ax\) (or by discrete analogues), the asymptotic stability of the system depends on the real parts of eigenvalues. Non-normality can nevertheless produce transient growth: the norm \(\|x(t)\|\) may increase substantially before eventual decay, even when all eigenvalues indicate stability. This is tied to the ability of non-orthogonal eigenvectors to reinforce energy temporarily. |
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4.1.2 Asymptotic versus short-term behavior
Because transient amplification is not captured by eigenvalues alone, stability in non-normal systems requires consideration of matrix functions such as \(e^{tA}\) and of quantities related to the resolvent. In numerical simulations, time discretization and perturbations can interact with this transient amplification, affecting observed stability margins.
4.2 Discretized differential operators
Non-normal matrices frequently arise from discretizing partial differential equations. When an operator includes directional transport, advection, or boundary-driven effects, discretizations often yield matrices with significant off-diagonal coupling. In such cases, the discrete system inherits non-normal features that manifest as spurious oscillations, overshoots, or rapid growth in intermediate steps.
4.3 Control theory applications
In linear control, one studies systems of the form \(\dot{x}=Ax+Bu\), with controllability and robustness influenced by the spectral and pseudospectral properties of \(A\). Non-normality can reduce robustness to disturbances and modeling errors, and it can change how feedback gains translate into performance because the system may amplify perturbations before feedback takes effect.
4.4 Fluid dynamics applications
Several fluid-flow models lead to non-normal operators, particularly those associated with shear flows and linearizations around nontrivial base states. Even when linear eigenvalues suggest decay, non-normality can enable transient amplification of perturbations, which is relevant to mechanisms such as growth of disturbances and the onset of turbulence in regimes where asymptotic behavior alone is misleading.
5 Measures and diagnostics of non-normality
5.1 Commutator norm
A direct diagnostic uses a norm of the commutator: \[
| \|AA^*-A^*A\|. |
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\] Different norms (e.g., Frobenius norm or spectral norm) emphasize different aspects of non-normality. While any such measure is not uniquely canonical, it provides a basis for comparing matrices and detecting strong non-normal effects.
5.2 Departure from normality
Because \([A,A^*]\) is zero exactly for normal matrices, its magnitude can be interpreted as the departure from normality. In practice, this departure may correlate with observed transient growth or with the widening of pseudospectra, though the relationship can be problem dependent.
5.3 Condition numbers of eigenvectors
| Another commonly used indicator is the conditioning of the eigenvector matrix for diagonalizable \(A\). If \(A\) is diagonalizable as \(A=V\Lambda V^{-1}\), then measures like \(\kappa(V)=\|V\|\|V^{-1}\|\) reflect how strongly non-orthogonal eigenvectors amplify perturbations. Large \(\kappa(V)\) often accompanies strong eigenvalue sensitivity. |
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5.4 Numerical indicators and tests
In computational practice, non-normality is often diagnosed indirectly through:
- large pseudospectral contours near the numerical region of interest;
| - unexpectedly large transient norms \(\|e^{tA}\|\) over moderate time windows; |
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- substantial differences between computed eigenvalues under perturbations or varying numerical precision;
| - large resolvent norms \(\|(zI-A)^{-1}\|\) in regions where \(z\) is not close to any eigenvalue. |
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6 Special classes and examples
6.1 Upper triangular matrices
Upper triangular matrices have eigenvalues equal to their diagonal entries, but the nilpotent or off-diagonal structure influences non-normal behavior. Many upper triangular matrices with nonzero entries above the diagonal are non-normal, and their departure from normality can be concentrated in specific superdiagonal patterns.
6.2 Nilpotent matrices
A matrix \(N\) is nilpotent if \(N^k=0\) for some \(k\). Nilpotent non-normal matrices (including Jordan blocks) illustrate how eigenvalues (which are all zero) fail to predict dynamic behavior: despite having spectrum at \(\{0\}\), such matrices can generate nontrivial polynomial growth in powers or in matrix exponentials.
6.3 Toeplitz and companion matrices
Certain structured matrices are frequently non-normal. Toeplitz matrices (constant diagonals) can be normal in special symmetric cases but typically are non-normal otherwise. Companion matrices of polynomials encode roots as eigenvalues; their non-normality affects the stability of root-finding and the sensitivity of polynomial root locations under perturbations.
6.4 Sparse non-normal matrices
In large-scale applications, one often works with sparse discretizations. Sparsity does not guarantee normality, and many sparse operators are strongly non-normal due to asymmetric couplings. This can complicate iterative methods for eigenvalues or linear solves, because convergence can depend on pseudospectral properties rather than just eigenvalues.
7 Theoretical results
7.1 Theorems on diagonalizability
Non-normality does not by itself determine whether a matrix is diagonalizable. Some non-normal matrices are diagonalizable but have non-orthogonal eigenvectors; others are not diagonalizable and exhibit Jordan blocks. Theory typically distinguishes between:
- normal matrices (unitarily diagonalizable),
- diagonalizable but non-normal matrices (similar to diagonal but with possibly ill-conditioned eigenbases),
- defective matrices (nontrivial Jordan blocks).
7.2 Bounds involving pseudospectra
Several results connect the growth of resolvent norms and pseudospectra to bounds on matrix functions, such as \(e^{tA}\) and \((zI-A)^{-1}\). These bounds formalize the idea that non-normality can create effective spectral regions where the system reacts strongly, even if eigenvalues are limited.
7.3 Relationships with matrix functions
| For analytic functions \(f(A)\), the operator norm \(\|f(A)\|\) is influenced by how \(f\) acts on the spectrum and by the non-orthogonality of eigenvectors and Jordan structure. In non-normal settings, classical eigenvalue-based intuition may underpredict \(\|f(A)\|\); instead, one often relies on resolvent estimates, contour integration representations, or pseudospectral bounds. |
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7.4 Invariant subspaces and triangularization
Non-normal matrices still admit structural decompositions. The Schur form yields invariant subspaces associated with the triangular structure of \(T\). Moreover, invariant subspaces related to Jordan chains or generalized eigenvectors govern how \(A\) acts on subspaces of the vector space, which in turn shapes transient amplification and the behavior of Krylov subspace methods.