Peskin & Schroeder about spontaneous symmetry breaking I am confused about how Peskin & Schroeder derive the matrix (21.39):
\begin{equation}
gF^a_{\ \ i}=\frac{v}{2} \left(\begin{array}{ccc}
g & 0 & 0 \\
0 & g & 0 \\
0 & 0 & g \\
0 & 0 & -g'
\end{array}\right),
\end{equation}
where $F^a_{\ \ i} \equiv T^a_{ij}\phi_{0j}$ ($T^a_{ij}$ are the real anti-symmetric generators of the group $SU(2) \times U(1)$, and $\phi_{0j}$ is the $j$-th vacuum expectation value). Just above for an example, they say that "$T^1 \phi_0$ equals $v/2$ times a unit vector in the $\phi_1$ direction", but according to their definition of $T^a$, one has $T^1=-i\frac{\sigma^1}{2}$ (where $\sigma^1$ is the first Pauli matrix), and so
\begin{equation}
T^1 \phi_0 = \frac{1}{\sqrt{2}}
\left(\begin{array}{cc}
0 & -i/2 \\
-i/2 & 0
\end{array}\right)
\left(\begin{array}{c}
0 \\
v 
\end{array}\right)
=
-\frac{1}{\sqrt{2}}\left(\begin{array}{c}
-iv/2 \\
0 
\end{array}\right),
\end{equation}
which is not normalized to $1$. Furthermore, this matrix is a column of two lines, and I don't understand how to recover their matrix (21.39) from two lines-vectors.
 A: This is for the expansion of the covariant derivative term $D_\mu\phi_i$ expanded near the vacuum state. The perturbation field $\phi-\phi_0=\frac{1}{\sqrt{2}}\left(\begin{array}{c}
\phi^{2}-i\phi^{1}\\
h+i\phi^{3}
\end{array}\right)$ is to be reorganized as four scalar fields $\chi=\left(\begin{array}{c}
\phi^1\\
\phi^2\\
\phi^3\\
h
\end{array}\right)$.
The terms in $\frac12(\partial_\mu \phi_i+gA^a_\mu T^a_{ij}\phi_j)^2$ include a first order perturbation term  $\partial^\mu \chi_i (g A^a_\mu T^a_{i0}\phi_0)$.
The massive Higgs boson field $h$ is always orthogonal to $F^a_i=gT^a_{i0}\phi_0$ so they turn the matrix to 4x3 instead of 4x4.
Your calculation of $T^1\phi_0$ is perfectly fine up to a sign. It's just that the resulting complex-2D field should be turned into the 3D real scalar field $\chi$. So $\partial \chi$ being paired to $\left(\begin{array}{c} vg/2\\ 0\\ 0 \end{array}\right)$ is to be treated as $\partial(\phi-\phi_0)$ being paired to $\frac{1}{\sqrt{2}}\left(\begin{array}{c} -ivg/2\\ 0\end{array}\right)$, which is the first row of the 4x3 matrix.
