Unitary Matrix

A unitary matrix is a square matrix with complex entries that preserves the standard inner product under matrix multiplication. Equivalently, its conjugate transpose equals its inverse. Unitary matrices are the complex counterparts of real orthogonal matrices and represent reversible, length-preserving linear transformations. They are fundamental in linear algebra, numerical computation, and quantum information. (quantum.cloud.ibm.com)

Definition and equivalent conditions

For an $n\times n$ complex matrix $U$, unitarity means

where $I_n$ is the identity matrix and $U^\dagger$ is the conjugate transpose:

This operation combines transposition with complex conjugation. Other common notations are $U^*$ and $U^H$. For square matrices, either of the two identities implies the other, and both say that the inverse matrix is $U^{-1}=U^\dagger$. (quantum.cloud.ibm.com)

Using the standard inner product $\langle x,y\rangle=x^\dagger y$, the defining condition gives

Consequently, $U$ preserves the Euclidean norm of every vector. Conversely, preservation of this norm for every vector implies unitarity. The condition concerns all vectors, not merely the individual coordinate vectors. (quantum.cloud.ibm.com)

Writing the columns as $u_1,\ldots,u_n$, the entries of $U^\dagger U$ are $\langle u_i,u_j\rangle$. Thus, the columns form an orthonormal basis of $\mathbb C^n$; the rows likewise form an orthonormal system. This identifies unitary matrices with changes between orthonormal coordinate systems. (ocw.mit.edu)

Geometric meaning and examples

As a linear map on a complex vector space, a unitary matrix is a bijective isometry. It preserves lengths, distances, and orthogonality without requiring the coordinates themselves to remain unchanged. In particular,

This follows directly by applying norm preservation to $x-y$. (quantum.cloud.ibm.com)

If all entries are real, conjugate transposition reduces to ordinary transposition. The unitary condition then becomes $U^TU=I$, precisely the definition of an orthogonal matrix. Plane rotations and reflections therefore provide real examples of unitary matrices. (netlib.org)

Simple complex examples are diagonal phase matrices:

Direct multiplication gives $D^\dagger D=I$, because each diagonal entry has modulus one. A familiar two-dimensional example is the Hadamard matrix

which satisfies $H^\dagger H=I$. Unlike a diagonal phase matrix, it mixes the input coordinates. (quantum.cloud.ibm.com)

Algebraic and spectral properties

Products and inverses of unitary matrices are unitary. For example,

The identity is also unitary, so these matrices form a group under multiplication, called the unitary group $U(n)$. Its determinant-one subgroup is denoted $SU(n)$. (math.mit.edu)

Taking determinants in $U^\dagger U=I$ gives

so $|\det U|=1$. This is a necessary condition, not a sufficient one: a diagonal matrix with entries $2$ and $1/2$ has determinant one but does not preserve lengths. These conclusions follow directly from the defining identity. (quantum.cloud.ibm.com)

Every unitary matrix is a normal matrix, since $U^\dagger U=UU^\dagger$. The spectral theorem therefore supplies a unitary diagonalization

Its eigenvalues all have modulus one: if $Ux=\lambda x$ and $x\ne0$, norm preservation yields $\|x\|=|\lambda|\|x\|$. Hence each eigenvalue can be written $e^{i\theta_j}$, and the action of $U$ in an orthonormal eigenbasis consists of independent phase multiplications. (ocw.mit.edu)

Numerical computation

Unitary transformations are important in numerical linear algebra because they do not amplify existing errors measured in the Euclidean norm. This supports numerical stability, although executing the transformations in floating-point arithmetic can still introduce rounding errors. (netlib.org)

In a full QR decomposition of a complex matrix, the factor $Q$ is square and unitary, while the other factor is triangular or trapezoidal. In the singular value decomposition,

both $P$ and $Q$ are unitary and $\Sigma$ contains nonnegative singular values. For a unitary matrix, all singular values equal one; consequently, its spectral-norm condition number is one. (netlib.org)

Quantum information and rectangular matrices

In finite-dimensional quantum mechanics, reversible evolution of an isolated system is represented by unitary matrices. Normalized state vectors remain normalized, while their amplitudes and measurement probabilities may change. A quantum gate acting on $k$ qubits is represented by a $2^k\times2^k$ unitary matrix. Measurement and general noisy processes are not, in general, described by a unitary matrix acting on the system alone. (quantum.cloud.ibm.com)

The square-matrix requirement distinguishes unitarity from a rectangular isometry. If an $m\times n$ matrix $Q$, with $m>n$, satisfies $Q^\dagger Q=I_n$, it preserves input norms but cannot satisfy $QQ^\dagger=I_m$. Instead, $QQ^\dagger$ is the orthogonal projection onto its column space. Such matrices occur as the reduced $Q$ factor in QR decompositions; they are not unitary in the strict square-matrix sense. (quantum.cloud.ibm.com)