= Gauge-boson mass matrix
{title2=$\mathcal L_{\mathrm{mass}}=\tfrac12 A_\mu^a(M^2)_{ab}A^{b\mu}$}
With canonically normalized <gauge field> <kinetic terms>, a gauge-boson mass matrix is the real symmetric matrix of coefficients in the displayed quadratic mass term. Its <eigenvalues> are squared physical <gauge boson> masses. For a complex <scalar field> vacuum $\Phi_0$ in the <Higgs mechanism>, $(M^2)_{ab}=2\operatorname{Re}[(g_aT_a\Phi_0)^\dagger(g_bT_b\Phi_0)]$. Thus for real $x_a$, $x^TM^2x=2\|\sum_a x_ag_aT_a\Phi_0\|^2\geq0$: it is a <Gram matrix> and a <positive semidefinite matrix>. An unbroken generator annihilates the <vacuum expectation value> and lies in its null space. Diagonalizing the matrix identifies the massive and massless field combinations.
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