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Let $A{v}_i={\lambda }_i{v}_i$ and $A{v}_j={\lambda }_j{v}_j$. The skew-Hermitian matrix property giveswhere the last equality uses part (a). Distinct eigenvalues therefore imply $\boxed{v_i^{†}{v}_j=0\text{ for }i\ne j}$. Normalize each eigenvector to length one. Since there are $n$ distinct eigenvalues, these vectors form an orthonormal basis.

Put them in the columns of $U$. Then $U$ is a unitary matrix and $AU=UD$, so

To avoid confusing the diagonalizing matrix with the exponential, write $A=WD{W}^{†}$ from part (b). The convergent matrix exponential series givesEach ${\lambda }_j$ is purely imaginary, so $|e^{{\lambda }_j}|=1$. The diagonal exponential is therefore a unitary matrix, andHence the matrix called $U=\exp (A)$ in this part is unitary. The series identity $(e^A{)}^{†}=e^{A^{†}}=e^{-A}$ gives the same conclusion directly and also covers repeated eigenvalues.

If $Av=\lambda v$, then $A^{n}v={\lambda }^{n}v$, and the matrix exponential series gives $Uv=e^{\lambda }v$. Also $(I+A{)}^{-1}v=(1+\lambda {)}^{-1}v$, since $1+\lambda \ne 0$. ThusThe assumed diagonalization ensures that these account for all eigenvalues, including multiplicities. In particular, for $\lambda =i\theta$,both of modulus one, in agreement with the unitary matrix property.

Here $A^2=-I$. Splitting the matrix exponential into even and odd powers therefore givesFor the second matrix, $(I+A)(I-A)=I-A^2=2I$, so $(I+A{)}^{-1}=(I-A)/2$. HenceBoth are real orthogonal matrices, and hence also unitary matrices.

The matrix sending the standard basis vectors to the given columns isWith the Hermitian inner product, the squared column norms are $3/6+1/6+2/6=1$, $3/6+1/6+2/6=1$, and $4/6+2/6=1$. The off-diagonal products areThus the columns form an orthonormal basis, so $\boxed{U^{†}U=I}$ and $U$ is a unitary matrix. Here the operation maps the old basis vectors to the new vectors; the inverse coordinate change of basis uses $U^{†}$.

Put $D=A-iB$, so $D^{†}=A+iB$. Because the Hermitian matrices $A,B$ commute,For every nonzero vector $v$,since $A$ is an invertible matrix. Thus $D$ and $D^{†}$ are invertible matrices, justifying the inverse in the definition of $U$. The conjugate transpose of $U=D^{†}{D}^{-1}$ is $(D^{†}{)}^{-1}D$. Since $D$ and $D^{†}$ commute, their inverses do too, andHence $U$ is a unitary matrix.

For the unitary matrix $C$, $C^{-1}=C^{†}$. Therefore $D=C^{-1}AC$ is a Hermitian matrix:Every positive integer power remains a Hermitian matrix, since $(D^p{)}^{†}=(D^{†}{)}^p=D^p$. Its eigenvalues are real by the finite-dimensional spectral theorem. Equivalently, matrix similarity giveswhose eigenvalues are $\boxed{{\lambda }_j^p\in \mathbb{R}}$, where the ${\lambda }_j$ are the real eigenvalues of $A$.

Write ${\gamma }^{\pm }=-i{\beta }^{\pm }$, $\overline{\beta }=({\beta }^{+}+{\beta }^{-})/2$ and $\chi =({\beta }^{+}-{\beta }^{-})z/2$. Then $\gamma z=-i\overline{\beta }z$ and $\delta z=-i\chi$. With $\cosh (-i\chi )=\cos \chi$ and $\sinh (-i\chi )=-i\sin \chi$, the corrected matrix becomesThe scalar is a common propagation wave phase and the real rotation matrix rotates the input linear polarization through $\chi$. There is no attenuation and no generated ellipticity: the field map is a unitary matrix. The two circular eigenmodes accumulate different wave phases, giving circular birefringence and optical activity.

The correct unfactored matrix remains finite where $\cos \chi =0$. In contrast, the printed $\mathrm{sech}$ form would give $e^{i\overline{\beta }z}{\sec }^2\chi$ times the rotation matrix, including unphysical divergent gain in this stipulated lossless case. This provides a direct physical check on the correction in part (d).

In a dense beam dump, long-lived pions and kaons usually interact and lose energy before decaying. Their residual stopped decays yield low-energy neutrinos, below the useful high-energy tau-interaction sample. Prompt charmed mesons have much shorter lifetimes and can decay before reinteraction. This suppresses the conventional muon neutrino background relative to the prompt component. The production sequence is Quantum chromodynamics creation of $c\bar{c}$, principally $gg\to c\bar{c}$ with $q\bar{q}\to g^{\ast }\to c\bar{c}$ also allowed, followed by hadronization $c\bar{s}\to D_s^{+}$ and its charge conjugate. The $gg$ process includes both quark-exchange diagrams and the three-gluon diagram. Fragmentation also produces other hadrons; the proton does not turn directly into one isolated Ds meson.The charge-conjugate chain gives a prompt tau antineutrino and a secondary tau neutrino. Thus each leptonic Ds meson chain produces both tau-flavour signs. The corresponding Feynman diagrams are shown in a.

For the CKM matrix, diagonalizing the up/down quark mass matrices by $U_{uL},U_{dL}$ leaves $V=U_{uL}^{†}{U}_{dL}$ in the charged current:The three-generation unitary matrix has three physical mixing angles and one CP violation phase after field rephasings. At fixed kinematics and hadronic matrix element, rates are proportional to $|V_{ij}{|}^2$; phase space, parent lifetimes, form factors and helicity suppression must also be included when comparing branching fractions.

For a charged pseudoscalar meson $P^{+}=(c\bar{q})$, leptonic pseudoscalar decay width givesThe Ds meson channel is Cabibbo-favoured, $|V_{cs}|\simeq 0.974$, whereas the charged D meson channel uses $|V_{cd}|\simeq 0.224$. Taking $m_{D_s}\simeq 1.968$, $m_D\simeq 1.870$ and $m_{\tau }\simeq 1.777$ in GeV givesIncluding the approximate lifetime ratio ${\tau }_{D_s}/{\tau }_D\simeq 0.48$ and the given production ratio $5.2/11$ gives a tau yield ratio about $17(f_{D_s}/f_D{)}^2$, already much greater than one even with equal decay constants. This is an illustrative estimate using approximate additional mass/lifetime inputs, not a determination from the production cross sections alone. A neutral D meson cannot annihilate to the charged $\tau \nu$ pair by electric charge conservation; its semileptonic tau modes have insufficient phase space once the required strange or nonstrange recoil hadron is included. Light mesons are below threshold; bottom production is much smaller at this fixed-target energy. Consequently neither the larger $D^0$ production rate nor ordinary light-hadron production outweighs the Ds meson tau source. ^[ref]

Strictly, it is the infinitesimal generators, not every unitary matrix with unit determinant, that are traceless. For a continuous one-parameter family $U(ϵ)=\exp (iϵT)$, unitarity requires the generator $T$ to be Hermitian. The determinant condition givesDifferentiating at $ϵ=0$ yields $\mathrm{tr}T=0$. A Hermitian matrix has $n$ real diagonal parameters and $n(n-1)$ real off-diagonal parameters; one real trace constraint leaves $n^2-1$ independent generators. ThusThe identity matrix supplies a concrete counterexample to interpreting the claim as one about arbitrary group elements: $I_n\in SU(n)$ but $\mathrm{tr}{I}_n=n$. Also, $e^{i\mathrm{tr}T}=1$ at a single parameter value only fixes the trace modulo $2\pi$; the continuous neighbourhood is essential.

The lepton mixing matrix relates charged-current neutrino flavour fields to mass eigenfields: ${\nu }_{\alpha L}={\sum }_i{U}_{\alpha i}{\nu }_{iL}$. Nondegenerate masses then give relative propagation phases and flavour oscillations. This is the three-massive-neutrino extension of the minimal massless-neutrino Standard Model.

A unitary three-by-three matrix has nine real parameters: three rotation angles and six phases. For Dirac leptons, the independent field rephasings remove five phases, because one common phase acts trivially. Three angles and one physical Dirac phase remain. For Majorana neutrinos two additional phases cannot be removed, but they cancel from ordinary oscillation probabilities.

The phases printed in the lower two rows do not give a unitary matrix for general $\delta$. A consistent convention is $U=R_{23}{U}_{13}(\delta )R_{12}$, where the real rotations use the indicated mixing angles andThe lower-row phase factors in the resulting full matrix are $e^{+i\delta }$, opposite to the $e^{-i\delta }$ in $U_{e3}$. For the source's all-minus assignment, the scalar product of the electron row with the conjugated muon row is $-2i{s}_{23}{s}_{13}{c}_{13}\sin \delta$, which is nonzero at $\delta =\pi /2$ and nonzero mixing. This supplies an explicit counterexample. The electron-row moduli used below are unaffected by this correction.^[ref]

The CKM matrix converts down-type quark mass eigenstates into the combinations entering the weak charged current. Its entry $V_{ij}$ controls a charged-current transition between up-type flavour $i$ and down-type flavour $j$. In the three-generation Standard Model, $V=U_{uL}^{†}{U}_{dL}$ is a product of unitary matrices from diagonalizing the quark mass matrices, so $V^{†}V=1$.

The Wolfenstein parametrization displays the hierarchyThus diagonal transitions dominate; first-second generation mixing is larger than second-third generation mixing, and direct first-third generation mixing is smallest. The parameters $A,\rho ,\eta$ refine this order estimate. With nondegenerate quark masses, a nonzero irreducible weak phase permits CP violation; at leading order the Jarlskog invariant is $J\simeq A^2{\lambda }^6\eta$. Individual complex entries are not enough: the phase must survive allowed quark-field rephasings.

The exact physical CKM matrix is unitary, whereas its displayed cubic truncation is only approximately so. For example its first-row norm is $1+{\lambda }^4/4+A^2{\lambda }^6({\rho }^2+{\eta }^2)$, which is not exactly one for nonzero $\lambda$. The omitted higher-order terms repair this; unitarity must be tested to the retained order.

A mass eigenstate $|{\nu }_i⟩$ has a definite neutrino mass $m_i$ and acquires a definite propagation phase at fixed momentum. An interaction eigenstate is the neutrino flavor selected by a weak charged current: ${\nu }_e$ is produced or detected together with an electron, and ${\nu }_{\mu }$ together with a muon. The two bases are related by a unitary matrix, here the real rotation matrix $U$.

Inverting the supplied transformation gives $|{\nu }_e⟩=c|{\nu }_1⟩+s|{\nu }_2⟩$ and $|{\nu }_{\mu }⟩=-s|{\nu }_1⟩+c|{\nu }_2⟩$, where $c=\cos \theta$, $s=\sin \theta$. If $\theta =0$, these bases coincide: each produced neutrino flavor propagates as one mass eigenstate, and there is no neutrino oscillation, even if the two masses differ. Conversely, equal masses also prevent vacuum neutrino oscillation, even when a nonzero basis-rotation angle is written down.

Combine the two equal-mass Dirac fields into $\Psi =({\psi }^1,{\psi }^2{)}^T$. A constant unitary matrix $U$ acting on this flavour index commutes with the common Dirac Hamiltonian, so $\Psi \mapsto U\Psi$ gives the unitary flavour symmetry of degenerate Dirac fields $U(2)$. Its real Lie algebra has dimension four. A convenient Hermitian generator basis is $T^0=1_2/2$, $T^a={\tau }^a/2$, $a=1,2,3$, where ${\tau }^a$ are the Pauli matrices.

For each generator the canonical anticommutation relations giveThe calculation is exactly the one in b, since $T^A$ commutes with both the Dirac gamma matrices and the common fermion mass matrix. ThusExplicitly their densities areThe two off-diagonal conserved currents are why counting only the individual fermion numbers misses two generators. The singlet generates $U(1)$; the other three generate special unitary group $SU(2)$. Unequal masses would break the off-diagonal conservation laws.

Scope of the count. Four is the count of currents of the displayed ${\bar{\psi }}^A{\gamma }^{\mu }{T}_{AB}{\psi }^B$ form. The completely free system also has an enhanced internal symmetry of free Dirac fields if mixing with charge-conjugate fields is admitted. For $m\ne 0$, write its two Dirac fields as four equal-mass Majorana spinors ${\chi }^I$. Their orthogonal group $O(4)$ rotations have six continuous generators, with conserved currents proportional to $i{\bar{\chi }}^I{\gamma }^{\mu }{\chi }^J$ for $I<J$; the four $U(2)$ currents form a subgroup. This larger count is relevant if “similar” means all local internal currents of the free system, without preserving a chosen fermion number. If $m=0$, the usual number-preserving chiral flavour symmetry of massless Dirac fields is $U(2{)}_L\times U(2{)}_R$ with eight currents; allowing charge conjugation mixing enlarges it to $U(4)$ with sixteen. These counts concern internal currents, excluding spacetime Noether currents.

In the stated ordered propagation basis, a downward input is transmitted downward and reflected upward, giving the first column $(1,1{)}^T/\sqrt{2}$. An upward input is reflected downward with the opposite reflection phase and transmitted upward, giving the second column $(-1,1{)}^T/\sqrt{2}$. ThereforeEqual mirror reflection phases exchange the two directions. Choosing a phase of $\pi$ per reflection givesEach path encounters one mirror, so $M$ describes both arms together, not two successive reflections of each photon. Changing its common phase leaves all detection probabilities unchanged. Direct multiplication gives $B^{†}B=M^{†}M=I$, so both are unitary matrices. They preserve the norm of a quantum state and hence total probability for these lossless optical elements.

Ideal detectors are represented by measurement matrices, but detection itself is not unitary evolution within the two traveling modes. With $e_1=(1,0{)}^T$ and $e_2=(0,1{)}^T$, the projective measurement operators areThey give Born rule probabilities $⟨\psi |P_j|\psi ⟩$ and sum to $I$. An absorbing detector maps the incident photon to an excited detector and vacuum field; a complete physical description includes those additional degrees of freedom.

For real angles write $u_p=\cos {\theta }_p$ and $v_p=\sin {\theta }_p$. For one independent pair $p>0$, the transformation and its conjugate readUsing the canonical anticommutation relations for the $\alpha$ modes givesThe remaining anticommutators vanish for independent momentum pairs. The displayed two-by-two matrix is unitary and its inverse is obtained by changing $i$ to $-i$.

When the same formula is indexed over both signs of momentum, its first row at $-p$ must agree with the conjugate of its second row at $p$. Thus a consistent global convention requiresThis is the odd-momentum fermionic Bogoliubov angle convention. If one checks only the first-row formulas, their cross anticommutator is $i\sin ({\theta }_p+{\theta }_{-p})$; its vanishing gives the condition modulo $\pi$, with possible signs absorbable in the $\alpha$ modes. Requiring both rows to describe the same operators fixes the convention above. Merely checking a separate unitary matrix at each momentum misses this redundancy: choosing ${\theta }_p={\theta }_{-p}=\pi /4$ in both first rows would give $\{c_p,c_{-p}\}=i$.

At $p=0$, the annihilation operator and its conjugate belong to one mode. The full displayed formula is consistent only with $v_0=0$; choose ${\theta }_0=0$ and ${\alpha }_0=c_0$. This mode is not a two-mode pair.

An optical element that adds a controlled wave phase to one path. On two path states, an ideal lossless phase shifter has the displayed unitary matrix. It changes their relative phase while leaving individual path probabilities unchanged; subsequent interference can turn that phase into an output-intensity change.

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.

The up-type and down-type fermion mass matrices need different left-handed transformations represented by unitary matrices to diagonalize them. Their relative transformation appears in the charged weak current as the Cabibbo-Kobayashi-Maskawa matrix. The weak neutral currents remain diagonal in the minimal Standard Model, since the same-charge couplings are generation-independent. Three nondegenerate generations give three mixing angles and one physical weak CP phase after quantum field rephasings.

For nondegenerate quark masses, changing the constant quantum phases of quark fields preserves their diagonal mass terms. Adopt the conventionSuch quantum field rephasings alter individual entries of the CKM matrix without altering physics. Physical CP symmetry in this mixing sector means that some allowed convention makes the CKM matrix real. Toshihide Maskawa emphasized the distinction between complex entries and an irremovable CP-violating phase.^[ref]

Here is a constructive two-generation argument. For a generic $2\times 2$ unitary matrix, use the three effective row/column phases to make $V_{11},V_{12},V_{22}$ real and positive. Column orthogonality then forces $V_{21}$ to be real and negative. Row normalization leaves just the real rotation specified by the Cabibbo angle. Zero entries are limiting cases and introduce no weak phase. Thus two-generation quark mixing cannot violate CP symmetry through this mechanism, even when its entries initially look complex.

For three quark generations, the Jarlskog invariant provides an obstruction to repeating that construction: all four phase factors cancel from its defining quartet. If its imaginary part is nonzero, no choice of quantum field rephasings can make every entry real. With nondegenerate quark masses, this proves physical CP violation in the mixing sector. This argument concerns the weak interaction; it does not dispose of the separate Strong CP problem.