# Unitary matrix

A **unitary matrix** is a complex square matrix whose conjugate transpose equals its inverse: a matrix \( U \) is unitary if \( U^{\dagger} = U^{-1} \), equivalently \( U^{\dagger}U = I = UU^{\dagger} \), where \( U^{\dagger} \) is obtained by transposing \( U \) and taking the complex conjugate of every entry.<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup><sup> • </sup><sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup> Unitary matrices are the complex analog of real orthogonal matrices: a real unitary matrix is exactly an orthogonal matrix.<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( U^{\dagger}U = I \), equivalently \( U^{\dagger} = U^{-1} \), for a square complex matrix<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup> |
| Columns and rows | The columns (and likewise the rows) of a unitary matrix form an orthonormal basis of \( \mathbb{C}^n \), with \( \langle v_i \mid v_j \rangle = \delta_{ij} \)<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup><sup> • </sup><sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup> |
| Length preservation | \( \|Ux\|_2 = \|x\|_2 \) for every vector \( x \); the 2-norm condition number of \( U \) is 1<sup>[4](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)</sup> |
| Eigenvalues | All eigenvalues have modulus 1 and can be written \( e^{i\alpha} \)<sup>[5](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)</sup> |
| Determinant | \( \lvert \det U \rvert = 1 \)<sup>[6](https://assets.lincolncountynevada.com/html/Handbook_EngMath_Matrix_Unitary.html)</sup> |
| Group structure | Products and inverses of unitary matrices are unitary; the \( n \times n \) unitary matrices form the group \( U(n) \)<sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup> |
| Spectral property | Every unitary matrix is normal and admits an orthonormal basis of eigenvectors<sup>[7](https://www.sciencedirect.com/topics/mathematics/unitary-matrix)</sup><sup> • </sup><sup>[5](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)</sup> |

## Equivalent characterizations

Several conditions are equivalent to unitarity, and each gives a different working picture of the same object.

**Inner-product preservation.** A matrix \( Q \in \mathbb{C}^{n \times n} \) is unitary if and only if the linear transformation it defines preserves the standard inner product: \( \langle Qx, Qy \rangle = \langle x, y \rangle \) for all \( x, y \). The proof is one line, \( (Qy)^*Qx = y^*Q^*Qx = y^*x \), and the converse also holds: any matrix that preserves length, \( \|Ax\|_2 = \|x\|_2 \) for all \( x \), is unitary.<sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup><sup> • </sup><sup>[4](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)</sup> Preserving the inner product means preserving both lengths and angles, so a unitary transformation is the complex version of a rigid motion: it can rotate or reflect, but it cannot stretch or shrink.<sup>[6](https://assets.lincolncountynevada.com/html/Handbook_EngMath_Matrix_Unitary.html)</sup>

**Orthonormal columns.** A matrix is unitary if and only if its columns form an orthonormal basis of \( \mathbb{C}^n \), and likewise for its rows.<sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup> This is often the quickest practical test: check that each column has norm 1 and that distinct columns are orthogonal under the Hermitian inner product.<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup>

**Squareness matters.** An orthonormal (semi-unitary) matrix need not be square; for example, a single column \( (\sqrt{2}/2, \sqrt{2}/2)^T \) satisfies \( Q^TQ = I \) but has no inverse. Only when an orthonormal matrix is square is it unitary, and then \( Q^{-1} = Q^H \) and \( QQ^H = I \) hold automatically.<sup>[4](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)</sup>

## Algebraic properties

**Closure.** If \( U_0 \) and \( U_1 \) are unitary, so is their product, since \( (U_0U_1)^H(U_0U_1) = U_1^H U_0^H U_0 U_1 = I \); the same holds for \( U^H \) and for \( U^{-1} = U^H \). The unitary \( n \times n \) matrices therefore form a group under multiplication, denoted \( U(n) \); the real orthogonal matrices form the subgroup \( O(n) \).<sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup><sup> • </sup><sup>[4](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)</sup> This closure matters numerically, because many algorithms apply a long sequence of unitary transformations and the product remains exactly unitary in exact arithmetic.<sup>[4](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)</sup>

**Normality and the spectral theorem.** From \( U^{\dagger} = U^{-1} \) it follows immediately that every unitary matrix commutes with its conjugate transpose, so every unitary matrix is a normal matrix.<sup>[7](https://www.sciencedirect.com/topics/mathematics/unitary-matrix)</sup> The spectral theorem then applies in a strong form: eigenvectors belonging to distinct eigenvalues are orthogonal, and it is always possible to find an orthonormal basis of eigenvectors for any unitary matrix, even when eigenvalues repeat.<sup>[5](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)</sup> In other words, a unitary matrix is unitarily diagonalizable, with its spectrum on the unit circle.<sup>[8](https://people.cs.uchicago.edu/~razborov/teaching/QuantumComputing/notes.pdf)</sup>

**Conditioning.** Because \( U \) and \( U^{-1} = U^H \) both have 2-norm equal to 1, the 2-norm condition number \( \kappa_2(U) = \|U\|_2 \, \|U^{-1}\|_2 \) equals 1. Multiplying by a unitary matrix therefore introduces no amplification of rounding error in the norm-wise sense, which is why numerical linear algebra builds algorithms around unitary transformations.<sup>[4](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)</sup>

## Eigenvalues and eigenvectors

The eigenvalue structure follows directly from the definition. If \( U|v\rangle = \lambda|v\rangle \), then combining this with its adjoint gives \( \langle v \mid v \rangle = |\lambda|^2 \langle v \mid v \rangle \), so \( |\lambda|^2 = 1 \). Every eigenvalue is unimodular and can be written \( e^{i\alpha} \) for some angle \( \alpha \).<sup>[5](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)</sup> The spectrum of a unitary matrix therefore lies on the unit circle in the complex plane, in contrast to Hermitian matrices, whose eigenvalues are real.<sup>[8](https://people.cs.uchicago.edu/~razborov/teaching/QuantumComputing/notes.pdf)</sup> Real eigenvalues are still possible, for example \( \pm 1 \); unitarity forces the modulus to be 1, not the value to be non-real.<sup>[5](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)</sup>

For eigenvectors, suppose \( U|v\rangle = e^{i\lambda}|v\rangle \) and \( U|w\rangle = e^{i\mu}|w\rangle \) with distinct eigenvalues. Then \( (e^{i\lambda} - e^{i\mu})\langle v \mid w \rangle = 0 \), so \( v \) and \( w \) are orthogonal. As with Hermitian matrices, the argument extends to repeated eigenvalues, so an orthonormal eigenbasis always exists.<sup>[5](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)</sup>

## Relation to other matrix classes

Unitary matrices sit inside the normal matrices and generalize several familiar real classes.<sup>[7](https://www.sciencedirect.com/topics/mathematics/unitary-matrix)</sup> A real unitary matrix is an orthogonal matrix, satisfying \( Q^TQ = I \).<sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup> Hermitian matrices, for which \( A^{\dagger} = A \), are a different family: for a unitary matrix the conjugate transpose is the inverse, while for a [Hermitian matrix](https://www.edgepedia.org/hermitian-matrix) it equals the matrix itself. The two classes overlap only in matrices whose eigenvalues are simultaneously real and of modulus 1, such as \( \pm I \) and the Householder reflections.<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup> Simple examples include rotation matrices \( \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \), which are orthogonal, and permutation matrices, which are real and unitary.<sup>[3](https://people.cas.uab.edu/~mosya/teaching/660.pdf)</sup>

## Role in quantum mechanics

In quantum mechanics, the state of a closed system evolves by a unitary operator: \( |\psi(t)\rangle = U(t)|\psi(0)\rangle \) with \( U^{\dagger}U = I \).<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup> Unitarity is what keeps the total probability of finding the system somewhere equal to 1 at all times, since the norm of the state vector is preserved.<sup>[9](https://www.feynmanlectures.caltech.edu/III_08.html)</sup> For a time-independent Hamiltonian \( H \), the propagator is \( U(t) = e^{-iHt/\hbar} \), and it is unitary because \( H \) is Hermitian.<sup>[2](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)</sup> Unitary transformations also serve as symmetry operators and as changes of orthonormal basis in quantum theory.<sup>[1](https://books.physics.oregonstate.edu/LinAlg/unitary.html)</sup>

Quantum computation uses the same structure: a quantum algorithm is a sequence of unitary transformations, called quantum gates, applied to a register of qubits. In 1995 Barenco and colleagues showed that one-qubit gates together with the controlled NOT gate form a universal set, using the [QR decomposition](https://www.edgepedia.org/qr-decomposition) of unitary matrices, and the cosine-sine decomposition gives an efficient way to implement general gates.<sup>[10](https://arxiv.org/abs/quant-ph/0504100)</sup>

## Computation and applications

Numerical software represents unitary matrices implicitly rather than storing them entry by entry. LAPACK, the standard linear algebra library, writes an orthogonal or unitary matrix \( Q \) as a product of elementary reflectors, \( Q = H_1 H_2 \cdots H_k \), where each Householder reflector has the form \( H = I - \tau vv^H \) with a scalar \( \tau \) and a vector \( v \). Routines beginning SORG- or CUNG- generate \( Q \) explicitly, while SORM- or CUNM- multiply by \( Q \) or \( Q^H \) without ever forming it.<sup>[11](https://www.netlib.org/lapack/lug/node128.html)</sup> The library's orthogonal and unitary factorization routines cover QR and LQ factorizations, the cosine-sine decomposition, Householder reflectors, and Givens rotations.<sup>[12](https://www.netlib.org/lapack/explore-html/d7/d69/group__unitary__top.html)</sup>

Unitary transforms also appear in signal processing. The multidimensional discrete [Fourier transform](https://www.edgepedia.org/fourier-transform) acts as a unitary transform between discrete periodic sequences on multidimensional lattices, a property that rests on the Hermitian orthogonality of the vectors defining the [DFT matrix](https://www.edgepedia.org/dft-matrix), and this is what makes the transform simple to invert with appropriate scaling.<sup>[13](https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/el.2012.3413)</sup>

## References

1. [Unitary Matrices, Section 4.5, Oregon State University Linear Algebra text](https://books.physics.oregonstate.edu/LinAlg/unitary.html)
2. [Lecture Notes, Chapter 6: Time Evolution in Quantum Mechanics, MIT OpenCourseWare 22.02](https://ocw.mit.edu/courses/22-02-introduction-to-applied-nuclear-physics-spring-2012/b5106a499ae03e36b5a2e002355668f9_MIT22_02S12_lec_ch6.pdf)
3. [Numerical Linear Algebra course notes, Section 2.5, University of Alabama at Birmingham](https://people.cas.uab.edu/~mosya/teaching/660.pdf)
4. [ALAFF, Subsection 2.2.4: Unitary matrices, University of Texas at Austin](https://www.cs.utexas.edu/~flame/laff/alaff/chapter02-unitary-matrices.html)
5. [Properties of Unitary Matrices, Section 4.4, Oregon State University Linear Algebra text](https://books.physics.oregonstate.edu/LinAlg/eigenunitary.html)
6. [Engineering Math handbook: Unitary Matrix](https://assets.lincolncountynevada.com/html/Handbook_EngMath_Matrix_Unitary.html)
7. [Unitary Matrix, ScienceDirect Topics (Andrilli & Hecker, Elementary Linear Algebra)](https://www.sciencedirect.com/topics/mathematics/unitary-matrix)
8. [Linear algebra notes for quantum computing, University of Chicago (Razborov)](https://people.cs.uchicago.edu/~razborov/teaching/QuantumComputing/notes.pdf)
9. [The Feynman Lectures on Physics, Vol. III Ch. 8: The Hamiltonian Matrix](https://www.feynmanlectures.caltech.edu/III_08.html)
10. [Decompositions of general quantum gates, arXiv:quant-ph/0504100](https://arxiv.org/abs/quant-ph/0504100)
11. [Representation of Orthogonal or Unitary Matrices, LAPACK Users' Guide](https://www.netlib.org/lapack/lug/node128.html)
12. [LAPACK: Orthogonal/unitary factors (QR, CS, etc.)](https://www.netlib.org/lapack/explore-html/d7/d69/group__unitary__top.html)
13. [Angeletti, P. (2013). Proof of unitarity of multidimensional discrete Fourier transform. Electronics Letters 49(7), 501-503.](https://ietresearch.onlinelibrary.wiley.com/doi/10.1049/el.2012.3413)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Linear and multilinear algebra › Matrix theory › Structured and special matrix classes*

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