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A matrix is Hermitian when $A^{†}=A$, and unitary when $U^{†}U=U{U}^{†}=I$.

If $A\mathbf{u}=\lambda \mathbf{u}$ with $\mathbf{u}\ne 0$, thenso every eigenvalue is real. If $A\mathbf{u}=\lambda \mathbf{u}$ and $A\mathbf{v}=\mu \mathbf{v}$, thenDistinct eigenvalues therefore have orthogonal eigenvectors.

The normalized eigenvectors form an orthonormal basis, so the matrix $U=({\mathbf{u}}_1\cdots {\mathbf{u}}_n)$ satisfies $U^{†}U=I$ and is unitary. Withthe equations $A{\mathbf{u}}_j={\lambda }_j{\mathbf{u}}_j$ say $AU=UD$, hence $A=UD{U}^{†}$. This is the finite-dimensional unitary diagonalization of a normal matrix.

A unitary conjugate of an arbitrary diagonal matrix need not be Hermitian: $U=(1)$ and $D=(i)$ give $UD{U}^{†}=(i)$. It is Hermitian exactly when all diagonal entries of $D$ are real.

For the displayed matrix, one valid decomposition isThe columns are orthonormal eigenvectors, so direct multiplication gives $UD{U}^{†}=A$.

A complex square matrix $A$ is a normal matrix if it commutes with its conjugate transpose:By unitary diagonalization of a normal matrix, there are a unitary matrix $U$ and a diagonal matrix $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_N)$ such thatThe matrix 2-norm is invariant under multiplication by unitary matrices, soFor a diagonal matrix,with equality on a coordinate vector belonging to an eigenvalue of greatest modulus. Hencethe matrix 2-norm of a normal matrix.

Biunitary transformations diagonalize the up- and down-type Yukawa matrices:The neutral Higgs couplings are then diagonal, but the charged weak current becomesA $3\times 3$ unitary matrix has nine real parameters. Rephasing the six quark mass eigenfields removes five phases because their common phase is baryon number. The Cabibbo-Kobayashi-Maskawa matrix therefore has four physical parameters: three mixing angles and one CP-violating phase.

Let $H_h$ be the periodic centered second-difference matrix minus the real diagonal matrix containing $V(x_m)$. It is a Hermitian matrix, so the semidiscrete system iswith a skew-Hermitian matrix generator. ConsequentlyIts exact propagator $e^{-it{H}_h}$ is a unitary matrix, and therefore the semidiscretization is stable in the discrete $2$-norm, uniformly for all times and mesh sizes.

Suppose first that the stated unitary and environment states exist. Preservation of the inner product givesThe environment states are normalized, so the Cauchy-Schwarz inequality gives $|⟨e_0|e_1⟩|\le 1$. Therefore

Conversely, writeand assume $|a|\le |b|$. If $b\ne 0$, put $c=a/b$, so $|c|\le 1$, and chooseThen $⟨e_0|e_1⟩=c$ and the desired output vectors have inner product $bc=a$, exactly matching the input vectors. If $b=0$, the inequality forces $a=0$, and one may take $|e_0⟩=|e_1⟩=|0⟩$.

In either case the input pair and output pair have the same Gram matrix. The isometry taking one pair to the other extends to a unitary matrix, so the required $U$ exists. Hencewhich is the environment-assisted two-state pure-state transformation criterion.

A complex matrix $U$ is unitary matrix whenIf $Uv=\lambda v$ with $v\ne 0$, unitarity preserves norms, soTherefore every eigenvalue satisfies

The periodic centered-difference kinetic matrix $T_h$ is real symmetric, and the sampled real potential $V_h$ is a real diagonal matrix. Therefore $-i{T}_h$ and $-i{V}_h$ are skew-Hermitian, so each matrix exponential in the Strang splitting is a unitary matrix. Their product is unitary as well. Hence

The continuous symmetry isStrictly, the diagonal central $U(1)$ acts trivially, so the faithful group is the quotient by that common center. Since $L$ and $R$ are unitary matrices,Every term is therefore unchanged by cyclicity of the matrix trace:and the same conjugation argument applies to $\mathrm{Tr}(M^{†}M)$ and $\mathrm{Tr}[(M^{†}M{)}^2]$.

Every invertible complex matrix has the unique polar decompositionwhere $P$ is a positive-definite Hermitian matrix and $U$ is a unitary matrix. Writing $P=e^B$ gives $B=\frac{1}{2}\log (M{M}^{\ast })$, which is Hermitian.

A complex matrix $A$ is skew-Hermitian when $A^{†}=-A$. Its matrix exponential $e^{tA}$ is unitary for every real $t$.

An SU(2) matrix is a two-by-two complex unitary matrix with determinant one. It has the form $\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$ with $|a{|}^2+|b{|}^2=1$. Its action on the Riemann sphere is a rotational Möbius transformation; opposite matrices induce the same sphere rotation.