Exercise 1.07
Imagine we have a tensor $X^{\mu\nu}$ and a vector $V^{\mu}$, with components
$$X^{\mu\nu}=\begin{pmatrix}
2 & 0 & 1 & -1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
-2 & 1 & 1 & -2
\end{pmatrix},
\quad \quad
V^{\mu}=(-1,2,0,-2)$$Find the components of:
(a) $X^\mu{}_\nu$
(b) $X_\mu{}^\nu$
(c) $X^{(\mu\nu)}$
(d) $X_{[\mu\nu]}$
(e) $X^\lambda{}_\lambda$
(f) $V^\mu V_\mu$
(g) $V_\mu X^{\mu\nu}$
(b)
$$
X_{\mu}{}^{\nu}=
\eta_{\mu\sigma} X^{\sigma \nu}
=\begin{pmatrix}
-2 & 0 & -1 & 1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
-2 & 1 & 1 & -2
\end{pmatrix}
$$
(c)
$$
\begin{align}
X^{(\mu \nu)}&=\frac{1}{2}(X^{\mu \nu}+X^{\nu \mu}) \\
&=\frac{1}{2}\begin{pmatrix}
2 & 0 & 1 & -1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
-2 & 1 & 1 & -2
\end{pmatrix}
+\frac{1}{2}\begin{pmatrix}
2 & -1 & -1 & -2 \\
0 & 0 & 1 & 1 \\
1 & 3 & 0 & 1 \\
-1 & 2 & 0 & -2
\end{pmatrix} \\
&=\frac{1}{2}\begin{pmatrix}
4 & -1 & 0 & -3 \\
-1 & 0 & 4 & 3 \\
0 & 4 & 0 & 1 \\
-3 & 3 & 1 & -4
\end{pmatrix}
\end{align}
$$
(d)
$$
\begin{align}
X_{[\mu \nu]}&=\frac{1}{2}(X_{\mu \nu}-X_{\nu \mu})=\frac{\eta_{\mu\sigma}\eta_{\nu\lambda}}{2}(X^{\sigma\lambda}-X^{\lambda\sigma}) \\
&=\frac{1}{2}\begin{pmatrix}
2 & 0 & -1 & 1 \\
1 & 0 & 3 & 2 \\
1 & 1 & 0 & 0 \\
2 & 1 & 1 & -2
\end{pmatrix}
-\frac{1}{2}\begin{pmatrix}
2 & 1 & 1 & 2 \\
0 & 0 & 1 & 1 \\
-1 & 3 & 0 & 1 \\
1 & 2 & 0 & -2
\end{pmatrix} \\
&=\frac{1}{2}\begin{pmatrix}
0 & -1 & -2 & -1 \\
1 & 0 & 2 & 1 \\
2 & -2 & 0 & -1 \\
1 & -1 & 1 & 0
\end{pmatrix}
\end{align}
$$
(e)
$$
X^{\lambda}{}_{\lambda}=\delta ^{\nu}_{\mu}X^{\mu}{}_{\nu}=\underbrace{ -2+0+0-2 }_\text{from (a)}=-4
$$
(f)
$$
V^{\mu}V_{\mu}=V^{\mu}\eta_{\mu \nu}V^{\nu}=1\cdot(-1)+2^{2}+0^{2}+(-2)^{2}=7
$$
(g)
$$
V_{\mu}X^{\mu \nu}=\eta_{\mu\sigma}V^{\sigma}X^{\mu \nu}
=\begin{pmatrix}
2 & 0 & 1 & -1 \\
-1 & 0 & 3 & 2 \\
-1 & 1 & 0 & 0 \\
-2 & 1 & 1 & -2
\end{pmatrix}\cdot\begin{pmatrix}
1 \\ 2 \\ 0 \\ -2
\end{pmatrix}
=
\begin{pmatrix}
4 \\ -5 \\ 1 \\ 4
\end{pmatrix}
$$
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