Introduction to Data Assimilation for Scientists and Engineers

• Transposition:

\(\displaystyle \left( \underline{\underline M} \; \underline{x}\right)^T = \underline{x}^T\, \underline{\underline M}^T\) and \(\displaystyle \left( \underline{\underline M} \; \underline{\underline H} \right)^T = \underline{\underline H}^T\, \underline{\underline M}^T\)

• Gradient computation:

  \(J(\underline{x})\)	\(\underline{\rm grad\ }J(\underline{x})\)
  \(\underline{u}^T \, \underline{x}\)	\(\underline{u}\)
  \(\underline{u}^T \, \underline{\underline M} \; \underline{x}\)	\(\underline{\underline M}^T\, \underline{u}\)
  \(\underline{x}^T \, \underline{\underline M} \; \underline{x}\)	\(\left(\underline{\underline M} +\,\underline{\underline M}^T \right)\, \underline{x}\)
  \(\underline{x}^T\,\underline{\underline H}^T\, \underline{\underline M} \; \underline{\underline H} \; \underline{x}\)	\(\underline{\underline H} \; \left(\underline{\underline M}+\underline{\underline M}^T\right) \, \underline{\underline H}^T \, \underline{x}\)