# Normal matrix

A **normal matrix** is a square matrix that commutes with its conjugate transpose, meaning it satisfies \( AA^{*} = A^{*}A \), where \( A^{*} \) denotes the transpose of \( A \) with each entry replaced by its complex conjugate.<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup> The property matters because of the spectral theorem: a complex square matrix is normal if and only if it can be diagonalized by a unitary matrix, that is, written as \( A = Q\Lambda Q^{H} \) with \( Q \) unitary and \( \Lambda \) diagonal.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> Normality is therefore the exact algebraic condition behind the existence of an orthonormal basis of eigenvectors.

| Key fact | Detail |
|---|---|
| Definition | \( AA^{*} = A^{*}A \), where \( A^{*} \) is the conjugate transpose<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup> |
| Spectral theorem | \( A \) is normal if and only if \( A = Q\Lambda Q^{H} \) for a unitary \( Q \) and diagonal \( \Lambda \)<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> |
| Equivalent test | \( \|Ax\|_2 = \|A^{*}x\|_2 \) for all \( x \in \mathbb{C}^n \)<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup> |
| Included families | Hermitian (\( A^{*} = A \)), skew-Hermitian (\( A^{*} = -A \)), unitary (\( AA^{*} = A^{*}A = I \)), and all diagonal matrices<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup><sup> • </sup><sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> |
| Eigenvalue locations | Real for Hermitian, purely imaginary for skew-Hermitian, on the unit circle for unitary<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup><sup> • </sup><sup>[3](https://www.sci.utah.edu/~akil/docs/courses/2020fall/math6610/lec08m.pdf)</sup> |
| Numerical meaning | Eigenvalues and eigenvectors of a normal matrix are well conditioned; nonnormal matrices can exhibit spectral instability<sup>[4](https://epubs.siam.org/doi/10.1137/1.9780898719673.ch10)</sup> |
| Standard example | \( \begin{pmatrix}1 & -2\\ 2 & 1\end{pmatrix} \) satisfies \( A^{T}A = AA^{T} = 5I \) but is neither symmetric nor orthogonal; its eigenvalues are \( 1 \pm 2i \)<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> |

## Definition and equivalent characterizations

The defining condition \( AA^{*} = A^{*}A \) is purely algebraic, but it has equivalent formulations that reveal its meaning. One is metric: \( A \) is normal if and only if \( \|Ax\|_2 = \|A^{*}x\|_2 \) for every vector \( x \in \mathbb{C}^n \), so \( A \) and \( A^{*} \) stretch every vector by the same amount.<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup> Another is spectral: normality holds exactly when \( A \) has a complete set of orthonormal eigenvectors.<sup>[5](https://doi.org/10.1090/noti2683)</sup>

Not every matrix is normal. The matrix \( \begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix} \) fails the condition, while \( \begin{bmatrix}1 & -1\\ 1 & 1\end{bmatrix} \) satisfies it.<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup> Normality is also stronger than diagonalizability: the matrix \( \begin{pmatrix}2 & 1\\ 0 & 1\end{pmatrix} \) has two distinct eigenvalues and is therefore diagonalizable, but \( B^{T}B \neq BB^{T} \), so it is not normal.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>

## Families of normal matrices

Several important matrix classes sit inside the normal set, which is one reason the concept is useful as a unifying umbrella.<sup>[1](https://foucart.github.io/teaching/notes/Matrix.pdf)</sup>

- **Hermitian matrices** satisfy \( A^{*} = A \); they are the complex analog of real symmetric matrices, and their eigenvalues are always real.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>
- **Skew-Hermitian matrices** satisfy \( A^{*} = -A \); they are normal because \( A(-A) = (-A)A \), and their eigenvalues are purely imaginary.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>
- **Unitary matrices** satisfy \( QQ^{H} = I \); they represent rotations and reflections in complex space, and their eigenvalues have absolute value 1, lying on the unit circle.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup><sup> • </sup><sup>[3](https://www.sci.utah.edu/~akil/docs/courses/2020fall/math6610/lec08m.pdf)</sup>
- **Diagonal matrices** are normal because a diagonal matrix commutes with its own conjugate transpose.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>
- **Circulant matrices**, whose diagonals wrap around cyclically, are normal and are diagonalized by the discrete [Fourier transform](https://www.edgepedia.org/fourier-transform) matrix.<sup>[6](https://mcube.lab.nycu.edu.tw/~cfung/docs/books/moon2000MMASP.pdf)</sup>

The taxonomy extends further in applications such as quantum information, where projections and density matrices form normal subclasses.<sup>[7](https://www.math.uwaterloo.ca/~jyard/qic710/F17/Qic710Lec9-2017.pdf)</sup>

## The spectral theorem

The central result about normal matrices is the spectral theorem: a complex square matrix \( A \) is unitarily diagonalizable if and only if it is normal.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> In the diagonalization \( A = Q\Lambda Q^{H} \), the columns of \( Q \) form an orthonormal basis of eigenvectors and the diagonal entries of \( \Lambda \) are the corresponding eigenvalues.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> Geometrically, the theorem guarantees a set of perpendicular axes, the eigenvectors, along which the transformation acts simply as a stretch or compression given by the eigenvalues.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>

The standard proof proceeds through the [Schur decomposition](https://www.edgepedia.org/schur-decomposition), which states that any square matrix can be written \( A = UTU^{*} \) with \( U \) unitary and \( T \) upper triangular.<sup>[8](https://www.ocf.berkeley.edu/~seanvernon/decal/notes/n4.pdf)</sup> If \( A \) is normal, this unitary change of basis preserves normality, so \( T \) is itself normal; comparing the top-left entries of \( T^{H}T \) and \( TT^{H} \) forces the off-diagonal entries of the first row to vanish, and an induction on the remaining block shows that a triangular normal matrix must be diagonal.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup> Conversely, if \( A = Q\Lambda Q^{H} \), then \( AA^{H} = Q\Lambda\Lambda^{H}Q^{H} \) and \( A^{H}A = Q\Lambda^{H}\Lambda Q^{H} \), and diagonal matrices commute, so \( A \) is normal.<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>

A consequence is that for a normal matrix, eigenvectors belonging to distinct eigenvalues are orthogonal, a stronger statement than the linear independence guaranteed for a general [diagonalizable matrix](https://www.edgepedia.org/diagonalizable-matrix).<sup>[2](https://ericdarve.github.io/NLA/content/normal_matrices.html)</sup>

## Real versus complex spectral theorems

The real and complex versions of the spectral theorem have different hypotheses. Over the complex numbers, the theorem covers the broad class of normal matrices; over the real numbers, it holds only for the narrower class of real symmetric matrices, which are orthogonally diagonalizable as \( A = Q\Lambda Q^{T} \) with real eigenvalues.<sup>[9](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web/inner-products-6.html)</sup> The gap exists because the characteristic polynomial must split into linear factors over the field, which the complex numbers always allow; a real orthogonal but non-symmetric matrix such as \( \begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix} \) has the characteristic polynomial \( x^2 + 1 \) with no real roots, so it cannot be real-diagonalized at all despite being normal.<sup>[9](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web/inner-products-6.html)</sup>

## Numerical significance

Normality controls how reliably eigenvalues can be computed and used. A normal matrix is the most general matrix whose Schur form is diagonal, and all of its eigenvalues and eigenvectors are well conditioned; a nonnormal matrix instead has a triangular Schur form, and its spectral representation can be unstable, with sensitivity governed by the condition of the Jordan basis.<sup>[4](https://epubs.siam.org/doi/10.1137/1.9780898719673.ch10)</sup> Departures from normality can grow exponentially with a problem parameter, for example in the nonnormal differentiation matrices of numerical analysis.<sup>[10](https://www.jstor.org/stable/j.ctvzxx9kj)</sup>

Two standard tools make the contrast precise. The \( \epsilon \)-pseudospectrum, the set of eigenvalues of all perturbations \( A + E \) with \( \|E\| < \epsilon \), consists for a normal matrix of disks of radius \( \epsilon \) about the eigenvalues, but for a highly nonnormal matrix it may be a much larger set.<sup>[5](https://doi.org/10.1090/noti2683)</sup> Similarly, the numerical range \( W(A) \), the set of values \( \langle Aq, q \rangle \) over unit vectors \( q \), equals the convex hull of the eigenvalues when \( A \) is normal, but can be a much larger convex set otherwise.<sup>[5](https://doi.org/10.1090/noti2683)</sup>

## Computation

Hermitian, skew-Hermitian, and unitary matrices, the main normal subclasses, admit condensed representations by few parameters that allow low-cost eigenvalue algorithms.<sup>[11](https://www.sciencedirect.com/science/article/pii/S0377042713004500)</sup> For general normal matrices, a 2024 method by Haoze He, a researcher in numerical mathematics, and Daniel Kressner, professor of computational mathematics at EPFL, called RandDiag, diagonalizes \( A \) by forming its Hermitian part \( H = (A + A^{*})/2 \) and skew-Hermitian part \( S = (A - A^{*})/2 \), drawing two independent standard normal random coefficients, and diagonalizing the [Hermitian matrix](https://www.edgepedia.org/hermitian-matrix) \( \mu_{H}H + \mu_{S}\,iS \) with a standard Hermitian eigensolver; the resulting unitary matrix diagonalizes \( A \) with probability one in exact arithmetic.<sup>[12](https://arxiv.org/abs/2405.18399)</sup> The approach reuses decades of mature Hermitian eigenvalue software such as LAPACK, and its off-diagonal error can also serve as a cheap test of whether a given matrix is close to normal.<sup>[12](https://arxiv.org/abs/2405.18399)</sup>

## References

1. [Lecture 2: Spectral Theorems, S. Foucart](https://foucart.github.io/teaching/notes/Matrix.pdf)
2. [Normal matrices, CME 302 Numerical Linear Algebra, E. Darve, Stanford](https://ericdarve.github.io/NLA/content/normal_matrices.html)
3. [The spectral theorem, MATH 6610, University of Utah](https://www.sci.utah.edu/~akil/docs/courses/2020fall/math6610/lec08m.pdf)
4. [Lectures on Finite Precision Computations, Ch. 10: Robustness to Nonnormality, SIAM](https://epubs.siam.org/doi/10.1137/1.9780898719673.ch10)
5. [AMS Notices: pseudospectra and the behavior of nonnormal matrices](https://doi.org/10.1090/noti2683)
6. [Circulant Matrices, Moon et al., 2000](https://mcube.lab.nycu.edu.tw/~cfung/docs/books/moon2000MMASP.pdf)
7. [Introduction to Quantum Information Processing, QIC 710, University of Waterloo](https://www.math.uwaterloo.ca/~jyard/qic710/F17/Qic710Lec9-2017.pdf)
8. [Special Classes of Matrices, lecture notes, UC Berkeley](https://www.ocf.berkeley.edu/~seanvernon/decal/notes/n4.pdf)
9. [Diagonalization of matrices in Inner Product Spaces, Dartmouth Linear Algebra Companion](https://math.dartmouth.edu/~trs/PreTeXtProjects/UGLinearAlgebraCompanion-CLI-v3/output/web/inner-products-6.html)
10. [Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press](https://www.jstor.org/stable/j.ctvzxx9kj)
11. [Computing eigenvalues of normal matrices via complex symmetric matrices, Journal of Computational and Applied Mathematics](https://www.sciencedirect.com/science/article/pii/S0377042713004500)
12. [A simple, randomized algorithm for diagonalizing normal matrices, H. He and D. Kressner, arXiv 2024](https://arxiv.org/abs/2405.18399)

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