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#+title:Matrix Properties
#+setupfile: ../../math_options.org
* Disclaimer
This site as of now just a technology demonstration and its claims
should not be taken as true (even though I myself am pretty confident
they are)
* Basic properties
\begin{align*}
A + B = B + A
\end{align*}
* Dot product
\begin{align*}
(A^\text{T})^\text{T} &= A\\
(A + B)^\text{T} &= A^\text{T} + B^\text{T}\\
(AB)^\text{T} &= B^\text{T}A^\text{T}
\end{align*}
* Transpose
\begin{align*}
\bm{a} \cdot \bm{b} &= \overline{\bm{b}} \cdot \bm{a}\\
\bm{a} \cdot \bm{b} &= \bm{a}^\text{T} \bm{b}
\end{align*}
* Hermitian transpose
\begin{align*}
A^\ast &= \left[\overline{a_{ij}}\right]\\
(\lambda A)^\ast &= \left[\overline{\lambda a_{ij}}\right] = \overline{\lambda} \left[a_{ij}\right]^\ast = \overline{\lambda}\,\overline{A}\\
A^\dag &= (A^\ast)^\text{T}\\
A^{\dag\dag} &= A
\end{align*}
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